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i am grateful to alekos kechris for informing me of t.dyck/ ; the proof given seems to be due to alain louveau .
i thank norm levenberg for references .
hough , j.b .
, krishnapur , m. , peres , y. , and virg , b. , _ zeros of gaussian analytic functions and determinantal point processes_. university lecture series , * 51*. american mathematical society , providence , ri , 2009 .
mester , p. , invariant monotone coupling need not exist .
* 41 * ( 2013 ) , 3a , 11801190 .
morris , b. , the components of the wired spanning forest are recurrent . _
probab . theory related fields _ * 125 * ( 2003 ) , 259265 .
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we describe the fundamental constructions and properties of determinantal probability measures and point processes , giving streamlined proofs .
we illustrate these with some important examples . we pose several general questions and conjectures .
primary 60k99 , 60g55 ; secondary 42c30 , 37a15 , 37a35 , 37a50 , 68u99 .
random matrices , eigenvalues , orthogonal projections , positive contractions , exterior algebra , stochastic domination , negative association , point processes , mixtures , spanning trees , orthogonal polynomials , completeness , bernoulli processes .
determinantal point processes were originally defined by macchi @xcite in physics . starting in the 1990s ,
determinantal probability began to flourish as examples appeared in numerous parts of mathematics @xcite .
recently , applications to machine learning have appeared @xcite .
a discrete determinantal probability measure is one whose elementary cylinder probabilities are given by determinants . more specifically ,
suppose that @xmath0 is a finite or countable set and that @xmath1 is an @xmath2 matrix .
for a subset @xmath3 , let @xmath4 denote the submatrix of @xmath1 whose rows and columns are indexed by @xmath5 .
if @xmath6 is a random subset of @xmath0 with the property that for all finite @xmath7 , we have e.dpm = ( qa ) , then we call @xmath8 a . the inclusion - exclusion principle in combination with yields the probability of each elementary cylinder event .
therefore , for every @xmath1 , there is at most one probability measure , to be denoted @xmath9 , on subsets of @xmath0 that satisfies .
conversely , it is known ( see , e.g. , b.lyons:det/ ) that there is a determinantal probability measure corresponding to @xmath1 if @xmath1 is the matrix of a positive contraction on @xmath10 ( in the standard orthonormal basis ) .
technicalities are required even to define the corresponding concept of determinantal point process for @xmath0 being euclidean space or a more general space .
we present a virtually complete development of their basic properties in a way that minimizes such technicalities by adapting the approach of b.lyons:det/ from the discrete case .
in addition , we use an idea of goldman b.goldman/ to deduce properties of the general case from corresponding properties in the discrete case .
space limitations prevent mention of most of what is known in determinantal probability theory , which pertains largely to the analysis of specific examples .
we focus instead on some of the basic properties that hold for all determinantal processes and on some intriguing open questions .
let @xmath0 be a denumerable set .
we identify a subset of @xmath0 with an element of @xmath11 in the usual way .
there are several approaches to prove the basic existence results and identities for determinantal probability measures .
we sketch the one used by b.lyons : det/. this depends on understanding first the case where @xmath1 is the matrix of an orthogonal projection .
it also relies on exterior algebra so that the existence becomes immediate .
any unit vector @xmath12 in a hilbert space with orthonormal basis @xmath0 gives a probability measure @xmath13 on @xmath0 , namely , @xmath14 associated to orthogonal projections @xmath15 .
we refer to b.lyons:det/ for details not given here .
identify @xmath0 with the standard orthonormal basis of the real or complex hilbert space @xmath10 . for @xmath16 ,
let @xmath17 denote a collection of ordered @xmath18-element subsets of @xmath0 such that each @xmath18-element subset of @xmath0 appears exactly once in @xmath17 in some ordering .
define @xmath19 if @xmath20 , then @xmath21 and @xmath22 .
we also define @xmath23 to be the scalar field , @xmath24 or @xmath25 .
the elements of @xmath26 are called of @xmath18 , or for short .
we then define the ( or ) of multivectors in the usual alternating multilinear way : @xmath27 for any permutation @xmath28 , and @xmath29 for any scalars @xmath30 ( @xmath31,\ ; e \in e'$ ] ) and any finite @xmath32 .
( thus , @xmath33 unless all @xmath34 are distinct . ) the inner product on @xmath26 satisfies e.ipdet = _ i , j when @xmath35 and @xmath36 are 1-vectors .
( this also shows that the inner product on @xmath26 does not depend on the choice of orthonormal basis of @xmath37 . )
we then define the ( or ) @xmath38 , where the summands are declared orthogonal , making it into a hilbert space .
( throughout the paper , @xmath39 is used to indicate the sum of orthogonal summands , or , if there are an infinite number of orthogonal summands , the closure of their sum . )
vectors @xmath40 are linearly independent iff @xmath41 . for a @xmath18-element subset @xmath3 with ordering @xmath42 in @xmath17 , write @xmath43 .
we also write @xmath44 for any function @xmath45 .
although there is an isometric isomorphism @xmath46 for @xmath47 , this does not simplify matters in the discrete case
. it will be very useful in the continuous case later , however .
if @xmath48 is a closed linear subspace of @xmath37 , written @xmath49 , then we identify @xmath50 with its inclusion in @xmath51 .
that is , @xmath52 is the closure of the linear span of the @xmath18-vectors @xmath53 . in particular ,
if @xmath54 , then @xmath55 is a 1-dimensional subspace of @xmath51 ; denote by @xmath56 a unit multivector in this subspace .
note that @xmath56 is unique up to a scalar factor of modulus 1 ; which scalar is chosen will not affect the definitions below .
we denote by @xmath15 the orthogonal projection onto @xmath48 for any @xmath49 or , more generally , @xmath57 .
l.projection for every closed subspace @xmath49 , every @xmath16 , and every @xmath58 , we have @xmath59 write @xmath60 and expand the product .
all terms but @xmath61 have a factor of @xmath62 in them , making them orthogonal to @xmath50 by e.ipdet/. a multivector is called or if it is the wedge product of 1-vectors .
b.whitney:book/ , p. 49 , shows that e.whitney .
we shall use the defined by duality : @xmath63 in particular , if @xmath64 and @xmath65 is a multivector that does not contain any term with @xmath66 in it ( that is , @xmath67 ) , then @xmath68 and @xmath69 . more generally , if @xmath70 with @xmath71 and @xmath72 , then @xmath73 and @xmath74 .
note that the interior product is sesquilinear , not bilinear , over @xmath25 .
for @xmath70 , write @xmath75 $ ] for the subspace of scalar multiples of @xmath12 in @xmath37 . if @xmath48 is a finite - dimensional subspace of @xmath37 and @xmath76 , then e.hwedge _ h
e = p_h^e_h + [ e ] ( up to signum ) . to see this ,
let @xmath77 be an orthonormal basis of @xmath48 , where @xmath78 .
put @xmath79 .
then @xmath80 is an orthonormal basis of @xmath81 $ ] , whence @xmath82 } = u_1 \wedge u_2 \wedge \cdots \wedge u_r \wedge v = \mv_h \wedge v = \mv_h \wedge e/\|p_h^\perp e\|\ ] ] since @xmath83 .
this shows e.hwedge/. similarly , if @xmath84 , then e.hvee _ h e = p_h e_h e^ ( up to signum ) .
indeed , put @xmath85 .
let @xmath86 be an orthonormal basis of @xmath48 with @xmath87 .
then @xmath88 ( up to signum ) , as desired .
finally , we claim that e.reverse = .
indeed , @xmath89 , so this is equivalent to @xmath90 thus , it suffices to show that @xmath91 by sesquilinearity , it suffices to show this for @xmath92 members of an orthonormal basis of @xmath48 .
but then it is obvious . for a more detailed presentation of exterior algebra ,
see b.whitney : book/. let @xmath48 be a subspace of @xmath37 of dimension @xmath93 . define the probability measure @xmath94 on subsets @xmath95 by e.xihpr ^h(\{b } ) : = ||^2 .
note that this is non-0 only for @xmath96 .
also , by l.projection/ , @xmath97 for @xmath96 , which is non-0 iff @xmath98 are linearly independent .
that is , @xmath99 iff the projections of the elements of @xmath100 form a basis of @xmath48
. let @xmath101 be any basis of @xmath48 .
if we use e.ipdet/ and the fact that @xmath102 for some scalar @xmath103 , then we obtain another formula for @xmath94 : we use @xmath104 to denote a random subset of @xmath0 arising from a probability measure @xmath94 . to see that e.dpm/ holds for the matrix of @xmath15 , observe that for @xmath96 , @xmath105 = \bigip{p_{\ext(h ) } \theta_b , \theta_b } = \bigip{\bigwedge_{e \in b } p_h e , \bigwedge_{e \in b } e } = \det [ \ip{p_h e , f}]_{e , f \in b}\ ] ] by e.ipdet/.
this shows that e.dpm/ holds for @xmath106 since @xmath107 @xmath94-a.s .
the general case is a consequence of multilinearity , which gives the following extension of e.dpm/. we use the convention that @xmath108 and @xmath109 for any multivector @xmath65 .
t.genprs if @xmath110 and @xmath111 are ( possibly empty ) subsets of a finite set @xmath0 , then e.genprs ^h[a_1 , a_2 = ] = . in particular , for every @xmath3
, we have e.included ^h[a ] = p_(h ) _ a^2 .
c.dualrep if @xmath0 is finite , then for every subspace @xmath49 , we have e.dualrep ^h^(\{e b } ) = ^h(\{b } ) .
these extend to infinite @xmath0 . in order to define @xmath94
when @xmath48 is infinite dimensional , we proceed by finite approximation .
let @xmath112 be infinite .
consider first a finite - dimensional subspace @xmath48 of @xmath37 .
define @xmath113 as the image of the orthogonal projection of @xmath48 onto the span of @xmath114 . by considering a basis of @xmath48 , we see that @xmath115 in the weak operator topology ( wot ) , i.e. , matrix - entrywise , as @xmath116 .
it is also easy to see that if @xmath117 , then @xmath118 for all large @xmath18 and , in fact , @xmath119 in the usual norm topology .
it follows that e.genprs/ holds for this subspace @xmath48 and for every finite @xmath120 .
now let @xmath48 be an infinite - dimensional closed subspace of @xmath37 .
choose finite - dimensional subspaces @xmath121 .
it is well known that @xmath115 ( wot ) . then e.detgenprs a ( p_h_k a ) ( p_h a ) , whence @xmath122 has a weak@xmath123 limit that we denote @xmath94 and that satisfies e.genprs/. we also note that for _ any _ sequence of subspaces @xmath113 , if @xmath124 ( wot ) , then @xmath125 weak@xmath123 because e.detgenprs/ then holds .
we call @xmath1 a if @xmath1 is a self - adjoint operator on @xmath37 such that for all @xmath126 , we have @xmath127 . a of @xmath1 is an orthogonal projection @xmath15 onto a closed subspace @xmath128 for some @xmath129 such that for all @xmath126 , we have @xmath130 , where we regard @xmath131 as the orthogonal sum @xmath132 . in this case , @xmath1 is also called the of @xmath15 to @xmath37 .
choose such a dilation ( see e.vecdilate/ or e.dilate/ ) and define @xmath9 as the law of @xmath133 when @xmath104 has the law @xmath94 .
then e.dpm/ for @xmath1 is a special case of e.dpm/ for @xmath15 .
of course , when @xmath1 is the orthogonal projection onto a subspace @xmath48 , then @xmath134 .
basic properties of @xmath9 follow from those for orthogonal projections , such as : t.q if @xmath1 is a positive contraction , then for all finite @xmath135 , e.qgenprs ^q= .
if e.dpm/ is given , then e.qgenprs/ can be deduced from e.dpm/ without using our general theory and , in fact , without assuming that the matrix @xmath1 is self - adjoint . indeed , suppose that @xmath136 is any diagonal matrix .
denote its @xmath137-entry by @xmath138 . comparing coefficients of @xmath138
shows that e.dpm/ implies , for finite @xmath3 , e.xe = ( ( q + x ) a ) . replacing @xmath5 by @xmath139 and
choosing @xmath140 gives e.qgenprs/. on the other hand , if we substitute @xmath141 , then we may rewrite e.xe/ as e.ze = ( ( q z + i - q ) a ) , where @xmath142 is the diagonal matrix of the variables @xmath143 .
let @xmath0 be finite .
write @xmath144 for @xmath3
. then e.ze/ is equivalent to e.affine _ a e ^q[= a ] z^a = ( i - q+qz ) .
this is the same as the laplace transform of @xmath9 after a trivial change of variables .
when @xmath145 , we can write @xmath146 with @xmath147 . thus , for all @xmath3 , we have a probability measure @xmath8 on @xmath148 is called if its generating polynomial @xmath149 z^a$ ] satisfies the inequality for all @xmath150 and all real @xmath151 .
this property is satisfied by every determinantal probability measure , as was shown by b.bbl:rayleigh/ , who demonstrated its usefulness in showing other properties , such as negative associations and preservation under symmetric exclusion processes . for a set @xmath152 ,
denote by @xmath153 the @xmath154-field of events that are measurable with respect to the events @xmath155 for @xmath156 .
define the @xmath154-field to be the intersection of @xmath157 over all finite @xmath158 .
we say that a measure @xmath8 on @xmath148 has if every event in the tail @xmath154-field has measure either 0 or 1 .
t.tail b.lyons:det/ if @xmath1 is a positive contraction , then @xmath9 has trivial tail . for finite @xmath0 and a positive contraction @xmath1 ,
define the of @xmath9 to be @xmath159 numerical calculation supports the following conjecture b.lyons:det/ : g.concave for all positive contractions @xmath160 and
@xmath161 , we have e.concave ( ( q_1+q_2)/2 ) ( ( q_1 ) + ( q_2))/2 . let @xmath0 be denumerable .
a function @xmath162 is called if for all @xmath163 and all @xmath164 , we have @xmath165 . an event is called increasing or if its indicator is increasing . given two probability measures @xmath166 , @xmath167 on @xmath148 , we say that and write @xmath168 if for all increasing events @xmath169 , we have @xmath170 .
this is equivalent to @xmath171 for all bounded increasing @xmath172 .
a of two probability measures @xmath166 , @xmath167 on @xmath148 is a probability measure @xmath173 on @xmath174 whose coordinate projections are @xmath166 , @xmath167 ; it is if @xmath175 by strassen s theorem @xcite , stochastic domination @xmath176 is equivalent to the existence of a monotone coupling of @xmath166 and @xmath167 .
t.dominate-infinite b.lyons:det/ if @xmath177 , then @xmath178
. it would be very interesting to find a natural or explicit monotone coupling .
a coupling @xmath173 has @xmath8 if for all events @xmath179 , we have @xmath180 .
q.unioncoupling @xcite given @xmath181 , is there a coupling of @xmath182 and @xmath183 with union marginal @xmath94 ? a positive answer is supported by some numerical calculation .
it is easily seen to hold when @xmath184 by c.dualrep/. in the sequel , we write @xmath185 if @xmath186 for all @xmath126 .
t.dominate @xcite if @xmath187 , then @xmath188 . by t.dominate-infinite/ , it suffices that there exist orthogonal projections @xmath189 and @xmath190 that are dilations of @xmath160 and @xmath161 such that @xmath191 .
this follows from namark s dilation theorem @xcite , which says that any measure whose values are positive operators , whose total mass is @xmath192 , and which is countably additive in the weak operator topology dilates to a spectral measure .
the measure in our case is defined on a 3-point space , with masses @xmath160 , @xmath193 , and @xmath194 , respectively . if we denote the respective dilations by @xmath195 , @xmath196 , and @xmath197 , then we set @xmath198 and @xmath199
. a positive answer in general to q.unioncoupling/ would give the following more general result by compression : if @xmath160 , @xmath161 and @xmath200 are positive contractions on @xmath37 , then there is a coupling of @xmath201 and @xmath202 with union marginal @xmath203
. it would be very useful to have additional sufficient conditions for stochastic domination : see the end of s.orthogpoly/ and g.fkdom/. for examples where more is known , see t.gmdom/. we shall say that the events in @xmath153 are @xmath158 and likewise for functions that are measurable with respect to @xmath153 .
we say that @xmath8 has if for every pair @xmath204 , @xmath205 of increasing functions that are measurable with respect to complementary subsets of @xmath0 , e.negass .
@xcite if @xmath206 , then @xmath9 has negative associations .
the details for finite @xmath0 were given in b.lyons : det/. for infinite @xmath0 , let @xmath204 and @xmath205 be increasing bounded functions measurable with respect to @xmath207 and @xmath208 , respectively . choose finite @xmath209 .
the conditional expectations @xmath210 $ ] and @xmath211 $ ] are increasing functions to which e.negass/ applies ( because restriction to @xmath212 corresponds to a compression of @xmath1 , which is a positive contraction ) and which , being martingales , converge to @xmath204 and @xmath205 in @xmath213 .
write @xmath214 for the distribution of a bernoulli random variable with expectation @xmath215 . for @xmath216 $ ] ,
let @xmath217 be the distribution of a sum of independent @xmath218 random variables . recall that @xmath75 $ ] is the set of scalar multiples of @xmath12 .
t.eigmix @xcite ; lemma 3.4 of @xcite ; ( 2.38 ) of @xcite ; @xcite let @xmath1 be a positive contraction with spectral decomposition @xmath219}$ ] , where @xmath220 are orthonormal .
let @xmath221 be independent .
let @xmath222 $ ] ; thus , @xmath223 . then @xmath224 .
hence , if @xmath225 , then @xmath226 . by t.dominate/
, it suffices to prove it when only finitely many @xmath227 . then by t.q/
, we have @xmath228 = \bigip{\bigwedge_{e \in a } q e , \theta_a } $ ] for all @xmath3 . now
@xmath229 } e & = \sum_{j \colon a \to \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e \\ & = \sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e\end{aligned}\ ] ] because @xmath230 and @xmath231 } e$ ] is a multiple of @xmath12 , so none of the terms where @xmath232 is not injective contribute .
thus , @xmath233 } e = \ebig{\sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } \\ & = \ebig{\sum_{j \colon a \to \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } = \be\bigwedge_{e \in a } \sum_k i_k p_{[v_k ] } e = \be \bigwedge_{e \in a } p_{\rh } e \,.\end{aligned}\ ] ] we conclude that @xmath234 = \be \leftip{\bigwedge_{e \in a } p_{\rh } e , \theta_a } = \ebig { \bp^{\rh}\left [ a \subseteq \ba \right ] } $ ] by e.qgenprs/. we sketch another proof : let @xmath235 be disjoint from @xmath0 with the same cardinality .
choose an orthonormal sequence @xmath236 in @xmath131 .
define then @xmath1 is the compression of @xmath15 to @xmath37 . expanding @xmath237 in the obvious way into orthogonal pieces and restricting to @xmath0 , we obtain the desired equation from e.xihpr/. the first proof shows more generally the following : let @xmath238 be a positive contraction .
let @xmath220 be ( not necessarily orthogonal ) vectors such that @xmath239 } \lloew i$ ] .
let @xmath240 be independent bernoulli random variables with @xmath241 .
write @xmath242}$ ]
. then @xmath243 .
this was observed by ghosh and krishnapur ( personal communication , 2014 )
. note that in the mixture of t.eigmix/ , the distribution of @xmath244 is determinantal corresponding to the diagonal matrix with diagonal @xmath245 .
thus , it is natural to wonder whether @xmath246 can be taken to be a general determinantal measure .
if such a mixture is not necessarily determinantal , must it be strongly rayleigh or at least have negative correlations ? here
, we say that a probability measure @xmath8 on @xmath148 has if for every pair @xmath5 , @xmath100 of finite disjoint subsets of @xmath0 , we have @xmath247 \le \bp [ a \subseteq \qba ] \bp [ b \subseteq \qba ] $ ] .
note that negative associations is stronger than negative correlations .
the most well - known example of a ( nontrivial discrete ) determinantal probability measure is that where @xmath6 is a uniformly chosen random spanning tree of a finite connected graph @xmath248 with @xmath249 . here ,
we regard a spanning tree as a set of edges .
the fact that holds for the uniform spanning tree is due to b.burpem/ and is called the transfer current theorem .
the case with @xmath250 was shown much earlier by b.kirchhoff/ , while the case with @xmath251 was first shown by b.bsst/. write @xmath252 for the uniform spanning tree measure on @xmath253 . to see that @xmath252 is indeed determinantal , consider the vertex - edge incidence matrix @xmath254 of @xmath253 , where each edge is oriented ( arbitrarily ) and the @xmath255-entry of @xmath254 equals 1 if @xmath256 is the head of @xmath66 , @xmath257 if @xmath256 is the tail of @xmath66 , and 0 otherwise .
identifying an edge with its corresponding column of @xmath254 , we find that a spanning tree is the same as a basis of the column space of @xmath254 .
given @xmath258 , define the at @xmath256 to be the @xmath256-row of @xmath254 , regarded as a vector @xmath259 in the row space , @xmath260 .
it is easy that the row - rank of @xmath254 is @xmath261 .
let @xmath262 and let @xmath65 be the wedge product ( in some order ) of the stars at all the vertices other than @xmath263 .
thus , @xmath264 for some @xmath265 .
since spanning trees are bases of the column space of @xmath254 , we have @xmath266 iff @xmath5 is a spanning tree .
that is , the only non - zero coefficients of @xmath65 are those in which choosing one edge in each @xmath259 for @xmath267 yields a spanning tree ; moreover , each spanning tree occurs exactly once since there is exactly one way to choose an edge incident to each @xmath267 to get a given spanning tree .
this means that its coefficient is @xmath268 .
hence , @xmath269 is indeed uniform on spanning trees .
simultaneously , this proves the matrix tree theorem that the number of spanning trees equals @xmath270_{x , y \ne x_0}$ ] , since this determinant is @xmath271
. one can define analogues of @xmath252 on infinite connected graphs @xcite by weak limits . for brevity ,
we simply define them here as determinantal probability measures .
again , all edges of @xmath253 are oriented arbitrarily .
we define @xmath272 as the closure of the linear span of the stars .
an element of @xmath273 that is finitely supported and orthogonal to @xmath272 is called a ; the closed linear span of the cycles is @xmath274 .
the is @xmath275 , while the is @xmath276 .
our discussion of the continuous " case includes the discrete case , but the discrete case has the more elementary formulations given earlier .
let @xmath0 be a measurable space .
as before , @xmath0 will play the role of the underlying set on which a point process forms a counting measure . while before we implicitly used counting measure on @xmath0 itself , now we shall have an arbitrary measure @xmath173 ; it need not be a probability measure . the case of lebesgue measure on euclidean space is a common one .
the hilbert spaces of interest will be @xmath277
. there may be no natural order in @xmath0 , so to define , e.g. , a probability measure on @xmath278 points of @xmath0 , it is natural to use a probability measure on @xmath279 that is symmetric under coordinate changes and that vanishes on the diagonal @xmath280 .
likewise , for exterior algebra , it is more convenient to identify @xmath281 with @xmath282 for @xmath283 .
thus , @xmath284 is identified with the function @xmath285_{i , j \in \{1 , \ldots , n\}}/\sqrt{n ! } $ ] .
note that @xmath286 \det [ { v_i(x_j ) } ] = \det [ u_i(x_j ) ] \det [ { v_i(x_j)}]^t \nonumber \\ & = \det [ u_i(x_j)][{v_i(x_j)}]^t = \det [ k(x_i , x_j)]_{i , j \in \{1 , \ldots , n\}}\end{aligned}\ ] ] with @xmath287 .
here , @xmath288 denotes transpose .
suppose from now on that @xmath0 is a locally compact polish space ( equivalently , a locally compact second countable hausdorff space ) .
let @xmath173 be a radon measure on @xmath0 , i.e. , a borel measure that is finite on compact sets .
let @xmath289 be the set of radon measures on @xmath0 with values in @xmath290 .
we give @xmath289 the vague topology generated by the maps @xmath291 for continuous @xmath172 with compact support ; then @xmath289 is polish .
the corresponding borel @xmath154-field of @xmath289 is generated by the maps @xmath292 for borel @xmath3 .
let @xmath293 be a simple point process on @xmath0 , i.e. , a random variable with values in @xmath289 such that @xmath294 for all @xmath295 .
the power @xmath296 lies in @xmath297 .
thus , @xmath298 $ ] is a borel measure on @xmath299 ; the part of it that is concentrated on @xmath300 is called the of @xmath293 .
if the intensity measure is absolutely continuous with respect to @xmath301 , then its radon - nikodym derivative @xmath302 is called the or the : since the intensity measure vanishes on the diagonal @xmath303 , we take @xmath302 to vanish on @xmath303
. we also take @xmath302 to be symmetric under permutations of coordinates .
intensity functions are the continuous analogue of the elementary probabilities e.dpm/. since the sets @xmath304 generate the @xmath154-field on @xmath300 for pairwise disjoint borel @xmath305 , a measurable function @xmath306 is the " @xmath18-point intensity function iff since @xmath293 is simple , @xmath307 , where @xmath308 . since @xmath302 vanishes on the diagonal , it follows from e.rn/ that for disjoint @xmath309 and non - negative @xmath310 summing to @xmath18 , again , this characterizes @xmath302 , even if we use only @xmath311 . in the special case that @xmath312 a.s .
for some @xmath313 , then the definition e.rn/ shows that a random ordering of the @xmath278 points of @xmath293 has density @xmath314 .
more generally , e.rn/ shows that for all @xmath315 , whence in this case , we call @xmath293 if for some measurable @xmath316 and all @xmath16 , @xmath317 @xmath301-a.e . here
, @xmath318 is the matrix @xmath319_{i , j \le k}$ ] . in this case
, we denote the law of @xmath293 by @xmath320 .
we consider only @xmath158 that are locally square integrable ( i.e. , @xmath321 is radon ) , are hermitian ( i.e. , @xmath322 for all @xmath323 ) , and are positive semidefinite ( i.e. , @xmath324 is positive semidefinite for all finite @xmath325 , written @xmath326 ) . in this case , @xmath158 defines a positive semidefinite integral operator @xmath327 on functions @xmath328 with compact support . for every borel @xmath3 , we denote by @xmath329 the measure @xmath173 restricted to borel subsets of @xmath5 and by @xmath330 the compression of @xmath158 to @xmath5 , i.e. , @xmath331 for @xmath332 .
the operator @xmath158 is locally trace - class , i.e. , for every compact @xmath3 , the compression @xmath330 is trace class , having a spectral decomposition @xmath333 , where @xmath334 are orthonormal eigenfunctions of @xmath330 with positive summable eigenvalues @xmath335 . if @xmath110 is the set where @xmath336 , then @xmath337 and @xmath338 converges on @xmath339 , with sum @xmath340-a.e .
equal to @xmath158 .
we normally redefine @xmath158 on a set of measure 0 to equal this sum .
such a @xmath158 defines a determinantal point process iff the integral operator @xmath158 extends to all of @xmath341 as a positive contraction @xcite .
the joint intensities determine uniquely the law of the point process ( * ? ? ?
* lemma 4.2.6 ) .
poisson processes are not determinantal processes , but when @xmath173 is continuous , they are distributional limits of determinantal processes . to see that a positive contraction defines a determinantal point process , we first consider @xmath158 that defines an orthogonal projection onto a finite - dimensional subspace , @xmath48
. then @xmath342 for every orthonormal basis @xmath343 of @xmath48 and @xmath344 is a unit multivector in the notation of s.ext/. because of e.prodtensor/ , we have i.e. , @xmath345/n!$ ] is a density with respect to @xmath346 .
although in the discrete case , the absolute squared coefficients of @xmath347 give the elementary probabilities , now coefficients are replaced by a function whose absolute square gives a probability density .
as noted already , e.firstdensity/ means that @xmath348 is the @xmath278-point intensity function . in order to show that this density gives a determinantal process with kernel @xmath158
, we use the cauchy - binet formula , which may be stated as follows : for @xmath349 matrices @xmath350 $ ] and @xmath351 $ ] with @xmath352_{\substack{i \le k \\ j \in j}}$ ] , we have @xmath353 [ b_{i , j}]^t\big ) = \sum_{|j| = k } \det a^j \cdot \det b^j = \sum _ { \substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k a_{i , \sigma(i ) } b_{i , \tau(i ) } \,,\ ] ] where @xmath354 denotes the image of @xmath154 and the sums extend over all pairs of injections @xmath355 here , the sign @xmath356 of @xmath154 is defined in the usual way by the parity of the number of pairs @xmath357 for which @xmath358 .
we have @xmath359 \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) \nonumber \\ & = \frac{1}{(n - k ) ! }
\int_{e^{n - k } } \sum_{\sigma \in \sym(n ) } ( -1)^\sigma \prod_{i=1}^n \phi_{\sigma(i)}(x_i ) \cdot { } \nonumber \\
\noalign{$\displaystyle \hfill \cdot \sum_{\tau \in \sym(n ) } ( -1)^\tau \prod_{i=1}^n \overline{\phi_{\tau(i)}(x_i ) } \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) $ } & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k \phi_{\sigma(i)}(x_i ) \overline{\phi_{\tau(i)}(x_i ) } \nonumber \\ & = \det \big(k \restrict ( x_1 , \ldots , x_k)\big ) \,.\end{aligned}\ ] ] here , the first equality uses e.integrate/ , the second equality uses e.prodtensor/ , the third equality uses the fact that @xmath360 is 1 or 0 according as @xmath361 or not , and the fourth equality uses cauchy - binet .
note that a factor of @xmath362 arises because for every pair of injections @xmath363 with equal image , there are @xmath362 extensions of them to permutations @xmath364 with @xmath361 for all @xmath365 ; in this case , @xmath366 .
we write @xmath94 for the law of the associated point process on @xmath0 .
l.weaklimit let @xmath367 with @xmath368 for some @xmath369 .
then @xmath370 is tight and every weak limit point of @xmath371 is simple . by using the kernel @xmath372 with respect to the measure @xmath373 , we may assume that @xmath374 .
tightness follows from @xmath375 \le \be[\sx_n(a ) ] = \int_a k_n(x , x ) \,d\mu(x)\,.\ ] ] for the rest , we may assume that @xmath0 is compact and @xmath376 .
let @xmath293 be a limit point of @xmath371 .
let @xmath377 be the atomic part of @xmath173 and @xmath378 .
choose @xmath379 and partition @xmath0 into sets @xmath380 with @xmath381 .
let @xmath5 be such that @xmath382 and @xmath383 .
let @xmath384 be open such that @xmath385 and @xmath386 .
then @xmath387 & \le \limsup_n \big(\bp[\sx_n(u \setminus a ) \ge 1 ] + \bp[\texists i \sx_n(a_i ) \ge 2]\big ) \\ & \le \limsup_n \big(\be[\sx_n(u \setminus a ) ] + \sum_i \be[(\sx_n(a_i))_2]\big ) \\ & \le \muc(u ) + \sum_i \mu(a_i)^2 < 2/m \ , .
\tag*{\qedhere}\end{aligned}\ ] ] now , given any locally trace - class orthogonal projection @xmath158 onto @xmath48 , choose finite - dimensional subspaces @xmath388 with corresponding projections @xmath389 .
clearly @xmath390 @xmath391-a.e.and @xmath392 @xmath173-a.e .
thus , the joint intensity functions converge a.e . by dominated convergence , if @xmath393 is relatively compact and borel , then @xmath394 \to \int_a \det ( k \restrict f ) \,d\mu^k(f)$ ] . by uniform exponential moments of @xmath395 (
* ? ? ?
* proof of lemma 4.2.6 ) , it follows that all weak limit points of @xmath396 are equal , and hence , by l.weaklimit/ , define @xmath94 with kernel @xmath158 .
( in s.cinequalities/ , we shall see that @xmath397 is stochastically increasing . ) finally , let @xmath158 be any locally trace - class positive contraction . define the orthogonal projection on @xmath398 whose block matrix is take an isometric isomorphism of @xmath277 to @xmath131 for some denumerable set @xmath235 and interpret the above as an orthogonal projection @xmath399 on @xmath400 .
then @xmath399 is clearly locally trace - class and @xmath158 is the compression of @xmath399 to @xmath0 .
thus , we define @xmath320 by intersecting samples of @xmath401 with @xmath0 .
we remark that by writing @xmath399 as a limit of increasing finite - rank projections that we then compress , we see that @xmath320 may be defined as a limit of determinantal processes corresponding to increasing finite - rank positive contractions .
g.ctail if @xmath158 is a locally trace - class positive contraction , then @xmath320 has trivial tail in that every event in @xmath402 is trivial . rather than using compressions as in the last paragraph above , an alternative approach to defining @xmath320 uses mixtures and starts from finite - rank projections , as in s.mix/. this approach is due to b.hkpv : survey/. consider first a finite - rank @xmath403 .
let @xmath404 be independent .
let @xmath405 $ ] ; thus , @xmath406 . we claim that @xmath407 is determinantal with kernel @xmath158 .
indeed , it is clearly a simple point process .
write @xmath408 , @xmath409 , and @xmath410 .
let @xmath411 .
combining cauchy - binet with e.prodtensor/ yields @xmath412 .
similarly , the joint intensities of @xmath413 are the expectations of the joint intensities of @xmath414 , which equal @xmath415 essentially the same works for trace - class @xmath416 ; we need merely take , in the last step , a limit in the above equation as @xmath417 for @xmath418 , since all terms are non - negative and @xmath419 a.e .
given this construction of @xmath320 for trace - class @xmath158 , one can then construct @xmath320 for a general locally trace - class positive contraction by defining its restriction to each relatively compact set @xmath5 via the trace - class compression @xmath330 . as noted by b.hkpv:survey/ ,
a consequence of the mixture representation is a clt due originally to b.soshnikov:gauss/ : t.clt let @xmath389 be trace - class positive contractions on spaces @xmath420 .
let @xmath367 and write @xmath421 . if @xmath422 as @xmath417 , then @xmath423 obeys a clt .
in order to simulate @xmath320 when @xmath158 is a trace - class positive contraction , it suffices , by taking a mixture as above , to see how to simulate @xmath424 when @xmath425 . the following algorithm ( * ? ? ? * algo .
18 ) gives a uniform random ordering of @xmath293 as @xmath426 . since @xmath427 , the measure @xmath428/n = n^{-1 } k(x , x)\,d\mu(x)$ ] is a probability measure on @xmath0 .
select a point @xmath429 at random from that measure .
if @xmath430 , then we are done .
if not , then let @xmath431 be the orthogonal complement in @xmath48 of the function @xmath432 , where @xmath343 is an orthonormal basis for @xmath48 .
then @xmath433 and we may repeat the above for @xmath431 to get the next point , @xmath434 , then @xmath435 , etc .
the conditional density of @xmath436 given @xmath437 is @xmath438 by e.densityktuple/ , i.e. , @xmath439 times the squared distance from @xmath440 to the linear span of @xmath441 .
it can help for rejection sampling to note that this is at most @xmath442 .
one can also sample faster by noting that the conditional distribution of @xmath436 is the same as that of @xmath443 , where @xmath444 is a uniformly random vector on the unit sphere of @xmath113 .
note that if @xmath445 are bounded @xmath446-valued random variables , then the function @xmath447 determines the joint distribution of @xmath448 since it gives the derivatives at @xmath449 of the probability generating function @xmath450 .
let us re - examine e.falling/ in the context of a finite - rank @xmath451 .
given disjoint @xmath452 and non - negative @xmath310 summing to @xmath18 , it will be convenient to write @xmath453 for @xmath454 .
we have by cauchy - binet @xmath455 = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \rho_k \,d\mu^k = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \det ( k \restrict ( x_1 , \ldots , x_k ) ) \,\prod_{j=1}^k d\mu(x_j ) \\ & = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,\prod_{j=1}^k d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \int_{a_{\kappa(j ) } } \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \lambda^{\operatorname{im}(\sigma ) } \prod_{j=1}^k \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_{\tau(j ) } } } \\ & = \sum_{\sigma \in \sym(k , n ) } ( -1)^\sigma \lambda^{\operatorname{im}(\sigma ) } \det \big [ \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_\ell } } \big]_{\substack{j \le k \hfill \\ \ell
\in \operatorname{im}(\sigma ) } } \,.\end{aligned}\ ] ] as an immediate consequence of this formula , we obtain the following important principle of goldman ( * ? ? ?
* proposition 12 ) that allows one to infer properties of continuous determinantal point processes from corresponding properties of discrete determinantal probability measures : t.transfer let @xmath456 and @xmath457 be two radon measure spaces on locally compact polish sets .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 and @xmath459 be pairwise disjoint borel subsets of @xmath325 .
let @xmath460 $ ] with @xmath461 .
let @xmath462 be orthonormal in @xmath277 and @xmath463 be orthonormal in @xmath464 .
let @xmath465 and @xmath466 . if @xmath467 for all @xmath468 , then the @xmath320-distribution of @xmath469 equals the @xmath470-distribution of @xmath471 .
when only finitely many @xmath227 , this follows from our previous calculation .
the general case follows from weak convergence of the processes corresponding to the partial sums , as in the paragraph following l.weaklimit/. this permits us to compare to discrete measures via ( * ? ? ?
* lemma 16 ) : l.compare let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 .
let @xmath472 for @xmath16 .
then there exists a denumerable set @xmath325 , pairwise disjoint subsets @xmath459 of @xmath325 , and @xmath473 such that @xmath474 and @xmath475 for all @xmath468 . without loss of generality
, we may assume that @xmath476 .
for each @xmath477 , fix an orthonormal basis @xmath478 for the subspace of @xmath277 spanned by @xmath479 . here ,
@xmath480 .
define @xmath481 and @xmath482 .
let @xmath483 be the isometric isomorphism from the span of @xmath484 to @xmath485 that sends @xmath486 to @xmath487 .
defining @xmath488 yields the desired vectors .
we now show how the discrete models of s.transf/ allow us to obtain the analogues of the stochastic inequalities known to hold for discrete determinantal probability measures . for a borel set @xmath7 , let @xmath207 denote the @xmath154-field on @xmath289 generated by the functions @xmath489 for borel @xmath490 .
we say that a function that is measurable with respect to @xmath207 is , more simply , measurable with respect to @xmath5 .
the obvious partial order on @xmath289 allows us to define what it means for a function @xmath491 to be . as in the discrete case
, we say that @xmath8 has if @xmath492 \le \be[f_1 ] \be[f_2 ] $ ] for every pair @xmath204 , @xmath205 of bounded increasing functions that are measurable with respect to complementary subsets of @xmath0 .
an event is increasing if its indicator is increasing .
then @xmath8 has negative associations iff for every pair @xmath493 , @xmath494 of increasing events that are measurable with respect to complementary subsets of @xmath0 .
we also say that and write @xmath495 if @xmath496 for every increasing event @xmath169 .
call an event if it has the form @xmath497 , where @xmath100 is a relatively compact borel set and @xmath498 .
write @xmath499 for the closure under finite unions and intersections of the collection of elementary increasing events with @xmath500 ; the notation @xmath501 is chosen for upwardly closed " .
note that every event in @xmath499 is measurable with respect to some finite collection of functions @xmath502 for pairwise _ disjoint _ relatively compact borel @xmath503 .
write @xmath504 for the closure of @xmath499 under monotone limits , i.e. , under unions of increasing sequences and under intersections of decreasing sequences ; these events are also increasing .
this is the same as the closure of @xmath499 under countable unions and intersections .
l.approxincr let @xmath5 be a borel subset of a locally compact polish space , @xmath0 .
then @xmath504 is exactly the class of increasing borel sets in @xmath207 .
we give a proof at the end of this subsection .
first , we derive two consequences . a weaker version ( negative correlations of elementary increasing events ) of the initial one is due to b.ghosh/. t.cfm let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath158 be a locally trace - class positive contraction on @xmath277 .
then @xmath320 has negative associations .
let @xmath505 be borel .
let @xmath506 and @xmath507 .
then @xmath508 for some compact @xmath100 by definition of @xmath509 .
we claim that e.cnegass/ holds for @xmath493 , @xmath494 , and @xmath510 , i.e. , for @xmath511 .
now @xmath493 is measurable with respect to a finite number of count functions @xmath502 for some disjoint @xmath512 ( @xmath513 ) and likewise @xmath494 is measurable with respect to a finite number of functions @xmath514 for some disjoint @xmath515 ( @xmath513 ) .
thus , there are functions @xmath516 and @xmath517 such that @xmath518 and @xmath519 . by t.transfer/ and l.compare/
, there is some discrete determinantal probability measure @xmath9 on some denumerable set @xmath325 and pairwise disjoint sets @xmath520 such that the joint @xmath521-distribution of all @xmath522 and @xmath523 is equal to the joint @xmath9-distribution of all @xmath524 and @xmath525 .
define the corresponding events @xmath526 by @xmath527 and @xmath528 .
since @xmath526 depend on disjoint subsets of @xmath325 , t.fm/ gives that @xmath529 .
this is the same as e.cnegass/ by t.transfer/. the same e.cnegass/ clearly then holds in the less restrictive setting @xmath530 by taking monotone limits .
l.approxincr/ completes the proof .
t.cdom theorem 3 of b.goldman/ suppose that @xmath531 and @xmath532 are two locally trace - class positive contractions such that @xmath533
. then @xmath534 .
it suffices to show that @xmath535 for every @xmath536 .
again , it suffices to assume that @xmath537 are trace class .
l.compare/ applied to all eigenfunctions of @xmath531 and @xmath532 yields a denumerable @xmath325 and two positive contractions @xmath538 on @xmath485 , together with an event @xmath539 , such that @xmath540 for @xmath541 .
furthermore , by construction , every function in @xmath485 is the image of a function in @xmath542 under the isometric isomorphism @xmath483 used to prove l.compare/ , whence @xmath543 .
therefore t.dominate/ yields @xmath544 , as desired .
again , it would be very interesting to have a natural monotone coupling of @xmath545 with @xmath546 .
for some examples where this would be desirable , see s.orthogpoly/. l.approxincr/ will follow from this folklore variant of a theorem of dyck b.dyck/ : t.dyck let @xmath136 be a polish space on which @xmath547 is a partial ordering that is closed in @xmath548 . let @xmath501 be a collection of open increasing sets that generates the borel subsets of @xmath136 .
let @xmath549 be the closure of @xmath501 under countable intersections and countable unions .
suppose that for all @xmath550 , either @xmath551 or there is @xmath552 and an open set @xmath553 such that @xmath554 , @xmath555 , and @xmath556 . then @xmath549 equals the class of increasing borel sets .
obviously every set in @xmath549 is borel and increasing . to show the converse
, we prove a variant of lusin s separation theorem .
namely , we show that if @xmath557 is increasing and analytic ( with respect to the paving of closed sets , as usual ) and if @xmath558 is analytic with @xmath559 , then there exists @xmath560 such that @xmath561 and @xmath562 . taking @xmath563 to be borel and @xmath564 forces @xmath565 and gives the desired conclusion . to prove this separation property , we first show a stronger conclusion in a special case
: suppose that @xmath566 are compact such that @xmath110 is contained in an increasing set @xmath563 that is disjoint from @xmath111 ; then there exists an open @xmath560 and an open @xmath567 such that @xmath568 , @xmath569 , and @xmath556 .
indeed , since @xmath563 is increasing , for every @xmath570 , we do _ not _ have that @xmath551 , whence by hypothesis , there exist @xmath571 and an open @xmath572 with @xmath573 , @xmath574 , and @xmath575 . because @xmath111 is compact , for each @xmath576 , we may choose @xmath577 such that @xmath578 .
define @xmath579 .
then @xmath580 is open , contains @xmath256 , and is disjoint from @xmath581 , whence compactness of @xmath110 ensures the existence of @xmath582 with @xmath583 .
then @xmath584 is open , contains @xmath111 , and is disjoint from @xmath384 , as desired . to prove the general case ,
let @xmath585 and @xmath586 be the two coordinate projections on @xmath587 .
define @xmath588 for @xmath589 to be 0 if there exists @xmath560 such that @xmath590 and @xmath591 ; and to be 1 otherwise .
we claim that @xmath192 is a capacity in the sense of ( * ? ? ?
* ( 30.1 ) ) .
it is obvious that @xmath592 if @xmath593 and it is simple to check that if @xmath594 , then @xmath595 .
suppose for the final property that @xmath5 is compact and @xmath596 ; we must find an open @xmath597 for which @xmath598 .
there exists some @xmath599 with @xmath600 and @xmath601 .
then the result of the second paragraph yields sets @xmath384 and @xmath567 that give @xmath602 as desired
. now let @xmath563 and @xmath603 be as in the first paragraph .
if @xmath604 is compact , then setting @xmath605 and applying the second paragraph shows that @xmath596 .
thus , by the choquet capacitability theorem ( * ? ? ?
* ( 30.13 ) ) , @xmath606 .
l.approxincr/ clearly every set in @xmath504 is increasing and in @xmath207 .
for the converse , endow @xmath5 with a metric so that it becomes locally compact polish while preserving its class of relatively compact sets and its borel @xmath154-field : choose a denumerable partition of @xmath5 into relatively compact sets @xmath607 and make each one compact and of diameter at most 1 ; make the distance between @xmath256 and @xmath608 be 1 if @xmath256 and @xmath608 belong to different @xmath607 .
let @xmath609 with the vague topology and let @xmath501 be the class of elementary increasing events defined with respect to ( relatively compact ) sets @xmath500 that are open for this new metric .
apply t.dyck/. since @xmath610 , the result follows .
natural examples of determinantal point processes arise from orthogonal polynomials with respect to a probability measure @xmath173 on @xmath25 .
assume that @xmath173 has infinite support and finite moments of all orders .
let @xmath389 denote the orthogonal projection of @xmath611 onto the linear span @xmath612 of the functions @xmath613 .
there exist unique ( up to signum ) polynomials @xmath614 of degree @xmath18 such that for every @xmath278 , @xmath615 is an orthonormal basis of @xmath612 . by elementary row operations
, we see that for variables @xmath616 , the map @xmath617_{i , j \le n}$ ] is a vandermonde polynomial up to a constant factor , whence @xmath618 [ \phi_i(z_j)]^ * = c_n \prod_{1 \le i < j \le n } |z_i - z_j|^2\ ] ] for some constant @xmath619 . therefore , the density of @xmath620 ( with points randomly ordered ) with respect to @xmath346 is given by @xmath621 times the square of a vandermonde determinant .
classical examples include the following : 1
. if @xmath173 is gaussian measure on @xmath24 , i.e. , @xmath622 , then @xmath614 are the hermite polynomials , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath624 , where @xmath625 is an @xmath626 matrix whose entries are independent standard complex gaussian .
( a standard complex gaussian random variable is the same as a standard gaussian vector in @xmath627 divided by @xmath628 in order that the complex variance equal 1 .
its density is @xmath629 with respect to lebesgue measure on @xmath25 . )
this is due to wigner ; see b.mehta/. 2 .
if @xmath173 is unit lebesgue measure on the unit circle @xmath630 , then @xmath631 , so @xmath632 , and @xmath620 is the law of the , which is the set of eigenvalues of a random matrix whose distribution is haar measure on the set of @xmath626 unitary matrices .
this ensemble was introduced by dyson , but the law of the eigenvalues is due to weyl ; see b.hkpv : book/. 3 .
if @xmath173 is standard gaussian measure on @xmath25 , then @xmath633 , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of an @xmath626 matrix whose entries are independent standard complex gaussian .
this is due to ginibre ; see b.hkpv : book/. 4 .
if @xmath173 is unit lebesgue measure on the unit disk @xmath634 , then @xmath635 , so @xmath636 , and the limit of @xmath620 is the law of the zero set of the random power series whose coefficients are independent standard complex gaussian , which converges in the unit disk a.s .
this is due to peres and virg b.peresvirag/. 5 .
if @xmath173 has density @xmath637 with respect to lebesgue measure on @xmath25 , then @xmath638 for @xmath315 , so @xmath639 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath640 when @xmath641 are independent @xmath626 matrices whose entries are independent standard complex gaussian .
( here , we are limited to @xmath612 since the larger spaces do not lie in @xmath341 . )
this is due to krishnapur b.krishnapur:thesis/ ; see b.hkpv : book/. the process was studied earlier by b.caillol/ and b.fjm/ , but without observing the connection to eigenvalues .
inverting stereographic projection , we identify this process with one whose density with respect to lebesgue measure on the unit sphere in @xmath642 is proportional to @xmath643 . for additional information on such processes ,
see @xcite . for an extension to complex manifolds ,
see @xcite . by t.cdom/ ,
the processes @xmath620 stochastically increase in @xmath278 for each of the examples above except the last
. it would be interesting to see natural monotone couplings .
perhaps the last example also increases stochastically in @xmath278 .
the is the limit of the @xmath278th ginibre processes as @xmath417 ; it has the kernel @xmath644 with respect to standard gaussian measure on @xmath25 .
this process is invariant under all isometries of @xmath25 . for each of the plane , sphere , and hyperbolic disk ,
there is only a 1-parameter family of determinantal point processes having a kernel @xmath645 that is holomorphic in @xmath646 and in @xmath647 and whose law is isometry invariant ( * ? ? ?
* theorem 3.0.5 ) . for the sphere , that family
has already been given above ; the parameter is a positive integer . for the other two families ,
the parameter is a positive real number , @xmath648 . in the case of the plane ,
the processes are related simply by homotheties , @xmath649 .
the push - forward of the ginibre process with respect to @xmath650 has kernel @xmath651 with respect to the measure @xmath652 , where @xmath173 is lebesgue measure on @xmath25 .
do these processes increase stochastically in @xmath648 , like poisson processes do ? in the hyperbolic disk , the processes have kernel @xmath653 with respect to the measure @xmath654 , where @xmath173 is lebesgue measure on @xmath655 .
( we fix a branch of @xmath656 for @xmath657 . )
these give orthogonal projections onto the generalized bergman spaces .
the case @xmath658 is that of the limiting ope4 above .
do these processes stochastically increase in @xmath648 ?
recall that when @xmath48 is a finite - dimensional subspace of @xmath37 , the measure @xmath94 is supported by those subsets @xmath95 that project to a basis of @xmath48 under @xmath15 .
similarly , when @xmath158 is the kernel of a finite - rank orthogonal projection onto @xmath659 , define the functions @xmath660 .
then the measure @xmath320 is supported by those @xmath661 such that @xmath662 is a basis of @xmath48 , since @xmath663 .
here , @xmath664 means that @xmath665 .
the question of extending this to infinite - dimensional @xmath48 turns out to be very interesting .
a basis of a finite - dimensional vector space is a minimal spanning set .
although @xmath666 is @xmath94-a.s .
linearly independent , minimality does not hold in general , even for the wired spanning forest of a tree , as shown by the examples in b.heicklenlyons/. see also c.ell2min/. however , the other half of being a basis does hold in the discrete case and is open in the continuous case .
let @xmath667 $ ] be the closed linear span of @xmath668 .
t.basis b.lyons:det/ for every @xmath49 , we have @xmath669 = h$ ] @xmath94-a.s .
we give an application of t.basis/ for @xmath670 , but it has an analogous statement for every countable abelian group . let @xmath671 be the unit circle equipped with unit lebesgue measure
. for a measurable function @xmath672 and @xmath673 , the of @xmath172 at @xmath278 is @xmath674 .
let @xmath675 denote the restriction of @xmath676 to @xmath677 .
if @xmath678 is measurable , we say @xmath679 is @xmath5 if the set @xmath680 is dense in @xmath681 , where we identify @xmath681 with the set of functions in @xmath682 that vanish outside @xmath5 .
the case where @xmath5 is an interval is quite classical ; see b.redheffer/ for a review . a crucial role in that case is played by the following notion of density of @xmath677 .
d.bm for an interval @xmath683 \subset \bbr \setminus \ { 0 \}$ ] , define its @xmath684\big ) : = \max \big\ { |a| , |b| \big\}/ \min \big\ { |a| , |b| \big\ } \,.\ ] ] for a discrete @xmath685 , the of @xmath677 , denoted @xmath686 , is the supremum of those @xmath687 for which there exist disjoint nonempty intervals @xmath688 with @xmath689 for all @xmath278 and @xmath690 ^ 2 = \infty$ ] .
a simpler form of the beurling - malliavin density was provided by b.red:two/ , who showed that e.bmred ( s ) = \ { c : s _ k s | - | < } .
c.seqdual b.lyons:det/ let @xmath691 be lebesgue measurable with measure @xmath692 .
then there is a set of beurling - malliavin density @xmath692 in @xmath693 that is complete for @xmath5 .
indeed , let @xmath694 be the determinantal probability measure on @xmath695 corresponding to the toeplitz matrix @xmath696
. then @xmath694-a.e .
@xmath697 is complete for @xmath5 and has @xmath698 .
when @xmath5 is an interval , the celebrated theorem of beurling and malliavin b.bm/ says that if @xmath677 is complete for @xmath5 , then @xmath699 , and that if @xmath700 , then @xmath677 is complete for @xmath5 .
( this holds for @xmath677 that are not necessarily sets of integers , but we are concerned in this subsection only with @xmath679 . )
c.seqdual/ can be compared ( take @xmath701 and @xmath702 ) to a theorem of bourgain and tzafriri b.btz/ , according to which there is a set @xmath697 of ( schnirelman ) density at least @xmath703 such that if @xmath704 and @xmath676 vanishes off @xmath677 , then @xmath705 it would be interesting to find a quantitative strengthening of c.seqdual/ that would encompass this theorem of @xcite .
the following theorem is equivalent to t.basis/ by duality : t.morris b.lyons:det/ for every @xmath49 , we have @xmath706 } h } = [ \ba]$ ] @xmath94-a.s . as an example , consider the wired spanning forest of a graph , @xmath253 . here , @xmath707 .
in this case , @xmath708 } \star(g ) } = \star(b)$ ] for @xmath709 .
thus , the conclusion of t.morris/ is that @xmath710 , which equals @xmath711 , is concentrated on the singleton @xmath712 for @xmath713-a.e .
@xmath714 .
this was a conjecture of , established by b.morris/. c.ell2min for every @xmath49 , @xmath94-a.s . the maps @xmath715 \to h$ ] and @xmath716 } \colon h \to [ \ba]$ ] are injective with dense image .
both statements are equivalent to @xmath717 \cap h^\perp = \{0\ } = h \cap \ba^\perp$ ] , and these are the contents of theorems [ thm : basis ] and [ thm : morris ] . proved that on any network @xmath718 ( where @xmath253 is the underlying graph and @xmath719 is the function assigning conductances , or weights , to the edges ) , for @xmath720-a.e .
forest @xmath714 and for every component tree @xmath483 of @xmath714 , the @xmath721 of @xmath722 equals @xmath483 a.s .
this suggested b.lyons:det/ the following extension .
given a subspace @xmath48 of @xmath37 and a set @xmath95 , the subspace of @xmath723 $ ] most like " or closest to " @xmath48 is the closure of the image of @xmath48 under the orthogonal projection @xmath724}$ ] ; we denote this subspace by @xmath725 .
for example , if @xmath726 , then @xmath727 since for each @xmath728 , we have @xmath724 } ( \star_x^g ) = \star_x^b$ ] .
to say that @xmath729 is concentrated on @xmath730 is the same as to say that @xmath731 $ ] .
this motivated the following theorem and shows how it is an extension of morris s theorem .
if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath732 , we have @xmath733 in other words , @xmath158 is a reproducing kernel for @xmath48 .
a subset @xmath677 of @xmath48 is called if the closed linear span of @xmath677 equals @xmath48 ; equivalently , the only element of @xmath48 that is orthogonal to @xmath677 is 0 .
an analogue of t.basis/ was conjectured by lyons and peres in 2010 : g.cbasis if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath320-a.e .
@xmath293 , @xmath734 = h$ ] , i.e. , if @xmath735 and @xmath736 , then @xmath737 .
just as in the discrete case , this appears to be on the critical border for many special instances , as we illustrate for several processes where @xmath738 : 1 .
let @xmath173 be lebesgue measure on @xmath24 and @xmath739 , the .
denote the fourier transform on @xmath24 by @xmath740 for @xmath741 , and , by isometric extension , for @xmath742 .
write @xmath743}$ ] . since @xmath744 , we have @xmath745 , where @xmath746 is the inverse fourier transform of @xmath172 .
therefore , the induced operator @xmath158 arises from the orthogonal projection onto the paley - wiener space @xmath747 .
the sine - kernel process arises frequently ; e.g. , it is various scaling limits of the @xmath278th gaussian unitary ensemble in the bulk " as @xmath417 .
( a related scaling limit of the gue is wigner s semicircle distribution . )
we may more easily interpret g.cbasis/ for fourier transforms of functions in @xmath748 $ ] : it says that for @xmath320-a.e .
@xmath293 , the only @xmath749 $ ] such that @xmath750 is @xmath751 .
although the beurling - malliavin theorem applies , no information can be deduced because @xmath752 a.s .
however , ghosh b.ghosh/ has proved this case .
2 . let @xmath173 be standard gaussian measure on @xmath25 and @xmath753 .
this is the ginibre process .
it corresponds to orthogonal projection onto the @xmath754 consisting of the entire functions that lie in @xmath611 ; this is the space of power series @xmath755 such that @xmath756 .
completeness of a set of elements @xmath757 in @xmath754 is equivalent to completeness in @xmath758 ( with lebesgue measure ) of the gabor system of windowed complex exponentials @xmath759 \st \lambda \in \sqrt{2}\lambda\big\ } \,,\ ] ] which is used in time - frequency analysis of non - band - limited signals
. the equivalence is proved using the bargmann transform @xmath760 \,dt \big ) \,,\ ] ] which is an isometry from @xmath758 to @xmath754 . that the critical density is 1 was shown in various senses going back to von neumann ; see b.clp/. this case has also been proved by ghosh b.ghosh/. 3 .
let @xmath173 be unit lebesgue measure on the unit disk @xmath761 and @xmath762 .
this process is the limiting ope4 in s.orthogpoly/. it corresponds to orthogonal projection onto the @xmath763 consisting of the analytic functions that lie in @xmath764 .
what is known about the zero sets of functions in the bergman space b.duren/ is insufficient to settle g.cbasis/ in this case and it remains open .
the two instances above that have been proved by ghosh b.ghosh/ follow from his more general result that g.cbasis/ holds whenever @xmath173 is continuous and @xmath320 is , which means that @xmath765 is measurable with respect to the @xmath320-completion of @xmath766 for every ball @xmath767 .
the limiting process ope4 is not rigid b.hs : tolerance/. ghosh and krishnapur ( personal communication , 2014 ) have shown that @xmath320 is rigid only if @xmath158 is an orthogonal projection .
it is not sufficient that @xmath158 be a projection , as the example of the bergman space shows .
a necessary and sufficient condition to be rigid is not known .
let @xmath158 be a locally trace - class orthogonal projection onto @xmath768 .
for a function @xmath172 , write @xmath769 for the function @xmath770
. let @xmath771 .
clearly @xmath772 for a.e .
@xmath293 . also , for @xmath773
, the function @xmath774 is bounded .
a conjecture analogous to c.ell2min/ is that @xmath293 is a sort of set of interpolation for @xmath48 in the sense that given any countable dense set @xmath775 , for a.e .
@xmath293 , the set @xmath776 is dense in @xmath777 .
one may also ask about completeness for appropriate poisson point processes .
suppose @xmath778 is a group that acts on @xmath0 and that @xmath158 is @xmath778-invariant , i.e. , @xmath779 for all @xmath780 , @xmath295 , and @xmath781 .
( this is equivalent to the operator @xmath158 being @xmath778-equivariant . )
then the probability measure @xmath320 is @xmath778-invariant .
this contact with ergodic theory and other areas of mathematics suggests many interesting questions .
lack of space prevents us from considering more than just a few aspects of the case where @xmath0 is discrete and from giving all definitions .
let @xmath782 . in this case
, @xmath158 is invariant iff @xmath783 for some @xmath784 $ ] , where @xmath785 .
we write @xmath786 in place of @xmath320 . some results and questions from b.ls:dyn/
follow .
t.bern for all @xmath172 , the process @xmath786 is isomorphic to a bernoulli process .
this was shown in dimension 1 by b.shitak:ii/ for those @xmath172 such that @xmath787 by showing that those @xmath786 are weak bernoulli ( wb ) , also called @xmath788-mixing " and absolutely regular " . despite its name
, it is known that wb is strictly stronger than bernoullicity . the precise class of @xmath172 for which @xmath786 is wb
is not known . as usual ,
the of a nonnegative function @xmath172 is @xmath789 .
t.gmdom for all @xmath172 , the process @xmath786 stochastically dominates product measure @xmath790 and is stochastically dominated by product measure @xmath791 .
these bounds are optimal .
we conjecture that ( kolmogorov - sinai ) entropy is concave , as would follow from g.concave/. g.invconcave for all @xmath172 and @xmath792 , we have @xmath793 .
q.block let @xmath794 $ ] be a trigonometric polynomial of degree @xmath795 .
then @xmath786 is @xmath795-dependent , as are all @xmath796-block factors of independent processes .
is @xmath797 an @xmath796-block factor of an i.i.d .
process ?
this is known when @xmath798 b.broman/. let @xmath778 be a sofic group , a class of groups that includes all finitely generated amenable groups and all finitely generated residually amenable groups .
no finitely generated group is known not to be sofic .
let @xmath0 be @xmath778 or , more generally , a set acted on by @xmath778 with finitely many orbits , such as the edges of a cayley graph of @xmath778 .
the following theorems are from b.lyonsthom/. t.sbern for every @xmath778-equivariant positive contraction @xmath1 on @xmath37 , the process @xmath9 is a @xmath799-limit of finitely dependent ( invariant ) processes . if @xmath778 is amenable and @xmath800 , then @xmath9 is isomorphic to a bernoulli process . even if @xmath166 and @xmath167 are @xmath778-invariant probability measures on @xmath801 with @xmath168 , there need not be a @xmath778-invariant monotone coupling of @xmath166 and @xmath167 b.mester : mono/. the proof of the preceding theorem depends on the next one : t.monojoin if @xmath160 and @xmath161 are two @xmath778-equivariant positive contractions on @xmath37 with @xmath185 , then there exists a @xmath778-invariant monotone coupling of @xmath201 and @xmath202 .
the proof of t.sbern/ also uses the inequality @xmath802 for equivariant positive contractions , @xmath1 and @xmath803 , where @xmath804 is the schatten 1-norm .
when @xmath1 and @xmath803 commute , one can improve this bound to @xmath805 we do not know whether this inequality always holds .
write @xmath806 for the fuglede - kadison determinant of @xmath1 when @xmath1 is a @xmath778-equivariant operator
. the following would extend t.gmdom/. it is open even for finite groups .
g.fkdom for all @xmath778-equivariant positive contractions @xmath1 on @xmath807 , the process @xmath9 stochastically dominates product measure @xmath808 and is stochastically dominated by product measure @xmath809 , and these bounds are optimal .
it turns out that the expected degree of a vertex in the free uniform spanning forest of a cayley graph depends only on the group , via its first @xmath810-betti number , @xmath811 , and not on the generating set used to define the cayley graph b.lyons:betti/ : t.betti in every cayley graph @xmath253 of a group @xmath778 , we have @xmath812 = 2 \beta_1(\gp ) + 2 \,.\ ] ] this is proved using the representation of @xmath813 as a determinantal probability measure .
it can be used to give a uniform bound on expansion constants b.lpv/ : t.lpv for every finite symmetric generating set @xmath677 of a group @xmath778 , we have @xmath814 for all finite non - empty @xmath815 .
there are extensions of these results to higher - dimensional cw - complexes and higher @xmath810-betti numbers b.lyons : betti/. in unpublished work with d. gaboriau @xcite , we have shown the following : t.damien let @xmath253 be a cayley graph of a finitely generated group @xmath778 and @xmath816 . then there exists a @xmath778-invariant finitely dependent determinantal probability measure @xmath9 on @xmath817 that stochastically dominates @xmath818 and such that @xmath819 \le \be_\fsf\big[\deg_\fo(\bp)\big ] + \epsilon \,.\ ] ] in addition , if @xmath778 is sofic , then @xmath820 .
if it could be shown that @xmath9 , or indeed every invariant finitely dependent probability measure that dominates @xmath813 , yields a connected subgraph a.s . , then it would follow that @xmath821 is equal to the cost of @xmath778 , a major open problem of b.gaboriau : invar/.
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in this appendix , we explain how the line width was extracted from the numerical data .
we begin by determining the spectral function , defined by @xmath115 this consists of a set of delta functions .
we then define the integrated spectral function @xmath116 .
this consists of a set of step functions ( see fig .
[ steps](a ) ) . for each step
, we identify the energy values corresponding to @xmath117 of the step , @xmath118 of the step , and @xmath119 of the step . the energy spacing between the @xmath117 and @xmath119 points
is taken to be the linewidth of this spectral line .
we track how this line width scales with @xmath0 .
we note that there is in general a wide distribution of line widths for any @xmath0 ( fig .
[ steps](b ) ) . as a result ,
the mean and the median linewidth scale very differently ( see fig.5 of the main text ) .
an understanding of the difference between the scaling of the mean and typical line width is an important challenge for future work .
( a ) the procedure for determining the linewidth .
the blue curve is an integrated spectral function .
the green squares divide each step into half , the red diamonds mark @xmath117 and the light blue circles mark @xmath120 of each step .
( b ) probability distribution of the linewidth @xmath109 for different values of coupling to the bath @xmath0 for a system with @xmath69 and @xmath121 averaged over 10 disorder configurations .
lines are a guide to the eye . ]
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we use exact diagonalization to study the breakdown of many - body localization in a strongly disordered and interacting system coupled to a thermalizing environment .
we show that the many - body level statistics cross over from poisson to goe , and the localized eigenstates thermalize , with the crossover coupling decreasing with the size of the bath in a manner consistent with the hypothesis that an infinitesimally small coupling to a thermodynamic bath should destroy localization of the eigenstates .
however , signatures of incomplete localization survive in spectral functions of local operators even when the coupling to the environment is sufficient to thermalize the eigenstates .
these include a discrete spectrum and a gap at zero frequency .
both features are washed out by line broadening as one increases the coupling to the bath .
we also determine how the line broadening scales with coupling to the bath .
isolated quantum systems with quenched disorder can enter a ` localized ' regime where they fail to ever reach thermodynamic equilibrium @xcite . while we have an essentially complete understanding of localization in non - interacting systems @xcite ,
the theory of many - body localization ( mbl ) is still under construction @xcite .
numerical investigations using exact diagonalization @xcite _ do _ indicate that all eigenstates of a strongly interacting disordered system can be localized .
most of the theoretical research so far has been in the limit of a perfectly isolated system .
however , experimental tests of mbl ( @xcite ) will always include some finite coupling to the environment .
what then can we expect to see in experiments designed to probe many body localization ?
a recent theory of mbl systems weakly coupled to heat baths proposed that while eigenstates are delocalized by an infinitesimally weak coupling to a heat bath , signatures of localization persist in spectral functions of local operators for weak coupling to a bath @xcite .
this theory has yet to face stringent numerical tests .
moreover , it did not discuss the spectral functions of the physical degrees of freedom , the quantities of direct relevance for experiments , focusing instead on the spectral functions of certain localized integrals of motion that are believed to exist @xcite , but which are related to the physical degrees of freedom by an unknown unitary transformation .
this work directly addresses these issues .
we use exact numerical diagonalization to establish the behavior of many body localized systems weakly coupled to heat baths .
we show that coupling @xmath0 to a bath results in a crossover from poisson to gaussian orthogonal ensemble ( goe ) eigenvalue statistics , which becomes exponentially steeper with increasing bath size . a similar rapid crossover to thermalization is seen in the eigenstates .
however , the prospect for seeing mbl in experiments is still realistic because signatures of incomplete localization remain in the spectral functions of local ( in real space ) operators .
indeed , we find that the spectral functions of the microscopic degrees of freedom look completely different in the localized and thermal phases ( see fig .
1 ) . the thermal phase has a continuous spectrum whereas the local spectral function in the localized phase is discrete , with a hierarchy of gaps , and a gap at zero frequency that survives even after spatial averaging .
increasing @xmath0 causes lines to broaden and fill in these gaps .
however , as long as the typical line broadening is less than the largest gaps , gap - like features remain .
our work also reveals how the line broadening scales with @xmath0 . _ the model _ : we choose the antiferromagnetic heisenberg spin-@xmath1 chain with random fields along @xmath2 : @xmath3 we set the interaction @xmath4 .
the on - site fields @xmath5 are independent random variables , uniformly distributed between @xmath6 and @xmath7 ; @xmath7 measures the disorder strength in the system .
this model with periodic boundary conditions has been shown to have a many - body localization transition at @xmath8 in the infinite temperature limit @xcite .
the hamiltonian in eq .
[ eq : h_pbit ] is written in terms of the physical degrees of freedom @xmath9 ( ` @xmath10-bits , ' in the language of @xcite , where @xmath10=physical ) .
in general , its eigenstates are quite complicated and non - trivial .
as shown @xcite , one can perform a unitary transformation to rewrite @xmath11 in terms of localized constants of motion @xmath12 .
the @xmath13 are dressed versions of the @xmath9 operators , which are localized in real space , with exponential tails , and are referred to in @xcite as ` @xmath14-bits ' ( @xmath14=localized )
. a unitary transformation to this ` @xmath14-bit ' basis can always be performed , if the system is in the regime where all the many body eigenstates are localized . in this @xmath14-bit basis
, the hamiltonian becomes @xmath15 the values of the coefficients @xmath16 and @xmath17 will depend upon the parent hamiltonian ( 1 ) , although these coefficients all fall off exponentially with distance .
the eigenstates of ( 2 ) are just products of @xmath18 .
motivated by the representation ( 2 ) of the hamiltonian ( 1 ) , it is instructive to consider the simpler hamiltonian @xmath19 where the @xmath20 and @xmath21 as independent random variables taken from a log - normal distribution with @xmath22 and @xmath23 , and similarly for @xmath24 .
we take @xmath25 and work with open - boundary conditions .
this hamiltonian also has the feature that eigenstates are product states of @xmath26 , and is simpler to work with numerically .
for the bath , we use a non - integrable hamiltonian that has been recently studied @xcite .
it consists of @xmath27 interacting spins with the hamiltonian : @xmath28 while using open boundary conditions , we add a boundary term @xmath29 to @xmath30 .
we use @xmath31 , @xmath32 and @xmath33 , values for which @xmath30 has been numerically shown by @xcite to have fast entanglement spreading .
( we use periodic boundary conditions only for @xmath10-bits with @xmath34 . ) the interaction between the system and bath should be local for both @xmath10- and @xmath14-bits .
we first study @xmath14-bit eigenstates , choosing the coupling : @xmath35 later we examine @xmath10-bit spectra , using the coupling @xmath36 the total hamiltonian is thus @xmath37 , where @xmath11 and @xmath38 are given by eq .
( 3 ) and ( 5 ) in the first part of this work , and by eq .
( 1 ) and ( 6 ) in the latter part of this work .
we will indicate the transition clearly in the text .
we use open boundary conditions except where periodic boundaries are explicitly mentioned .
we start by analyzing the breakdown of localization when the @xmath14-bit hamiltonian ( 3 ) is coupled to a bath according to ( 5 ) , by examining the many - body eigenvalue statistics as @xmath0 is increased from @xmath39 .
we perform exact diagonalization on a system with @xmath40 spins coupled to @xmath41 spins in the bath .
the many body level - spacing is @xmath42 , where @xmath43 is the energy of the @xmath44th eigenstate .
following @xcite , we define the ratio of adjacent gaps as @xmath45 .
we average this over eigenstates and several different realizations of the disorder to get a probability distribution @xmath46 at a particular value of @xmath0 . in fig .
[ fig : level_space ] , we show how @xmath46 evolves from poisson to goe like as @xmath0 is increased . in a localized system
we expect that @xmath47 , and for a thermalizing system , we expect that @xmath48 .
-bit hamiltonian as @xmath0 is increased .
results are for a system with @xmath40 spins and bath with @xmath41 spins averaged over @xmath49 eigenstates obtained from several disorder configurations .
the dark blue solid line is the poisson distribution expected for localized systems , and the light blue dashed line is the goe distribution expected for thermalizing systems . ]
the transition from poisson to goe statistics happens gradually for this finite size system . a simple analytical estimate of the characteristic value of @xmath0 at the crossover point proceeds as follows ( see also @xcite ) : if @xmath50 is the bandwidth of the bath and @xmath51 is the many body level spacing in the bath , then the system couples to @xmath52 states , with a typical matrix element to each state of order @xmath53 .
the coupling to the bath will be effective in thermalizing the system when this matrix element becomes of order the level spacing in the bath , i.e. when @xmath54 .
this indicates that the crossover coupling @xmath55 . since @xmath56
, the critical value of @xmath0 is expected to scale as @xmath57 . to quantitatively compare this crossover estimate to the data ,
we define @xmath58 . after averaging over disorder distributions
, @xmath59 should be @xmath60 in the goe regime and @xmath61 in the localized regime @xcite .
it is convenient to define the normalized quantity @xmath62=(@xmath63 , such that @xmath64 if the level statistics are goe and @xmath65 if they are poisson .
fig .
3(a ) shows how @xmath66 varies with @xmath0 for systems of size @xmath67 .
fig .
[ fig : r_trans](b ) shows that scaling of the form @xmath68 is successful in making the data for different @xmath27 in fig .
[ fig : r_trans](a ) collapse onto one curve .
data collapse occurs also for @xmath69 and @xmath70 , indicating clearly that it is @xmath27 which controls the finite size scaling .
we get the best collapse when the constant in the exponential is @xmath71 which is in good agreement with the analytical estimate @xmath72 .
this implies that the crossover to thermalization is at a coupling @xmath73 that is exponentially small in system size , so that level statistics become goe at infinitesimal @xmath0 in the thermodynamic limit .
( defined in the text ) in the @xmath14-bit hamiltonian as @xmath0 is increased for system sizes @xmath74 and @xmath75 .
data is averaged over @xmath49 eigenstates obtained from several disorder configurations .
( b ) collapse of data in ( a ) is in good agreement with analytic arguments for the finite size scaling presented in the main text , and depends only on @xmath27 . ]
another test of thermalization is checking whether the eigenstates obey the eigenstate thermalization hypothesis ( eth ) @xcite . the eth states that the expectation value of a local operator should be the same in every eigenstate within a small energy window . for a localized system
this will not be the case . in fig .
[ fig : eth ] , we show how eigenstate thermalization sets in as @xmath0 is increased .
we choose an energy window around the center of the band and calculate the standard deviation of the expectation value of @xmath76 for all eigenstates within the window .
explicitly , we define @xmath77,\ ] ] where the overline denotes averaging over an energy window of width @xmath78 in the middle of the band and @xmath79 is an eigenstate of the coupled system and bath .
we choose @xmath80 . after averaging over disorder distributions ,
we expect to find @xmath81 for a thermalized system .
fig .
[ fig : eth](a ) shows how @xmath82 approaches 0 as @xmath0 is increased for different system sizes . fig .
[ fig : eth](b ) shows that @xmath82 scales with @xmath0 similar to @xmath66 .
the exponent here is @xmath83 , also close to the estimated analytical value .
-bit hamiltonian as @xmath0 is increased for system sizes @xmath69 , @xmath84 , @xmath85 , @xmath86 , @xmath87 and @xmath75 .
@xmath82 as defined in the text is measured at the site of the central spin .
data is averaged over @xmath49 eigenstates obtained from several disorder configurations .
( b ) collapse of data in ( a ) agrees with analytical estimates of finite size scaling for @xmath88 . for a finite size system with @xmath27 spins in the bath ,
the eigenstates become effectively thermal for @xmath89 , implying that eigenstates in the thermodynamic limit become thermal for infinitesimal @xmath0 . ]
we now turn to an analysis of the spectral functions of local operators .
henceforth we are working with the physical degrees of freedom , eq .
( 1 ) and ( 6 ) .
we examine the spectral function from an exact eigenstate @xmath90 where @xmath91 is the @xmath92 eigenstate of the combined system and bath .
we note that since we are working with a finite size system with a discrete spectrum , the spectral function will always consist of a set of delta functions . at @xmath93 ,
the delta functions should have minimum spacing @xmath94 , equal to the many body level spacing in the system . at non - zero @xmath0 , each ` parent ' delta function will split into exponentially many descendants , with a typical spacing @xmath95 . a fine binning in energy with bin size greater than @xmath95
will then yield a smooth spectral function , with the ` parent ' delta functions of the system having been ` broadened ' by coupling to the bath . to investigate this broadening ,
it is convenient to take @xmath96 .
we therefore take @xmath97 and @xmath98 , and investigate how the ` line broadening ' evolves with @xmath0 for @xmath99 .
details of the procedure are outlined in the supplementary material , and the results are illustrated in fig .
[ fig : linewidth ] for @xmath100 .
the mean and median linewidth at a particular value of @xmath0 are significantly different .
this is a result of the long tails in the distribution of the linewidth ( see supplement ) .
fig .
[ fig : linewidth ] shows that at the larger values of @xmath0 we study , a log - log plot of the median vs @xmath0 appears to fit well to a straight ( dashed ) line . for
the system sizes that we are able to access , the straight line fit suggests @xmath101 , where @xmath102 increases as the size of the bath increases , reaching @xmath103 for @xmath104 .
we note that while a simple application of the golden rule predicts @xmath105 , a more careful analysis @xcite suggests that the true scaling should be @xmath106 .
the solid lines in fig .
[ fig : linewidth ] are a fit to this theoretical prediction , and are consistent with the data , except at smallest @xmath0 .
the discrepancy at smallest @xmath0 and the difference between median and mean are worthwhile topics for future work . for a system of @xmath10-bits with @xmath69 and
@xmath107 averaged over more than 38000 eigenstates obtained from several disorder configurations at @xmath100 .
@xmath108 for the sizes shown here .
the mean and the median of the probability distribution of the linewidth @xmath109 are extracted from the data as discussed in the appendix .
the dotted lines are linear fits to the data .
the solid lines are fits to the theoretical prediction .
[ fig : linewidth ] ] finally , we analyze the behavior of the spectral function averaged over all sites and eigenstates of the system , for @xmath110 .
we note that the hamiltonian ( 1 ) has a delocalization - localization phase transition at @xmath8 .
fig .
[ fig : pbits](a ) shows @xmath111 on the delocalized side of the transition for a small value of @xmath0 .
@xmath111 is smooth everywhere .
( the graininess is a result of the small system size . )
fig .
[ fig : pbits](b ) is on the localized side of the transition , with the system almost decoupled from the bath . here
, @xmath111 consists of clusters of narrow spectral lines , with a hierarchy of energy gaps , just as was shown to be the case for @xmath14-bit spectral functions in @xcite .
@xmath111 vanishes at @xmath112 .
thus , local spectral functions can distinguish between extended and localized phases . in fig .
[ fig : pbits](c - e ) we examine how the @xmath10-bit spectral functions evolve as @xmath0 increases .
we see that the line broadening increases and different lines start to overlap with each other , washing out the weaker spectral features , but larger gaps remain .
the zero - frequency gap also fills in with increasing @xmath0 .
the spectral functions retain signatures of localization even for @xmath113 when the eigenstates of the combined system and bath are effectively thermal , and get washed out when @xmath0 becomes comparable to the characteristic energy scales in the system ( i.e. @xmath114 ) . in conclusion
, we have investigated the signatures of localization in a disordered system weakly coupled to a heat bath using exact diagonalization .
the wave functions are found to exhibit a crossover to thermalization as a function of coupling to the bath .
the crossover coupling is proportional to the many body level spacing in the bath , and vanishes exponentially fast in the limit of a large bath size .
in contrast , the spectral functions of local operators are found to show more robust signatures of proximity to a localized phase .
while the spectral functions are smooth and continuous in the delocalized phase ( after coarse graining on the scale of the many body level spacing ) , the spectral functions in the localized phase consist of narrow spectral lines , and contain a hierarchy of gaps , as well as a gap at zero frequency that persists even after spatial averaging .
increasing the coupling to the bath increases the line broadening ( in a manner that we calculate ) and washes out these features .
however , signatures of localization survive in the spectral functions even at couplings to the bath where the exact eigenstates are effectively thermal ( fig .
1 ) . _
acknowledgments : _ rn would like to thank sarang gopalakrishnan and david huse for a collaboration on related ideas .
this work was supported by doe grant de - sc0002140 .
rnb . acknowledges the hospitality of the institute for advanced study , princeton while this work was being done .
rn was supported by a pcts fellowship .
sj was supported by the porter ogden jacobus fellowship of princeton university .
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in this supplementary material , we compare the value of the penetration depth obtained from experiments @xcite with the prediction from homes law ; for the latter , we use a combination of the experimental data obtained from optical - conductivity and dc transport . for each value of the doping ( @xmath8 )
, we estimate the ( approximate ) dc resistivity ( @xmath80 ) by extrapolating the curves to @xmath9 , from the transport data in fig.1(b ) of ref.@xcite .
we estimate the value of @xmath81 , where @xmath23 is the superconducting gap , from the data for optical conductivity in the superconducting state , as shown in fig .
3(b ) of ref . @xcite . since @xmath7 remains relatively unchanged as a function of @xmath8 in the vicinity of optimal doping , we assume @xmath82 to be independent of @xmath8 such that @xmath83@xmath84s@xmath85 .
then , in the dirty limit , _ s = _ . in order to obtain the penetration depth
, we need to restore various dimensionful constants such that , _
l^2(0)= , where @xmath86 m / s ) is the speed of light and @xmath87 f / m ; 1 f=1 @xmath88s ) is the permitivity of free space .
the values obtained are shown in the table below and have been presented in fig . 2 of the main text , along with a comparison to the experimental data @xcite .
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we present a theory for the large suppression of the superfluid - density , @xmath0 , in bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of a putative spin - density wave quantum critical point at a p - doping , @xmath4 .
we argue that the transition becomes weakly first - order in the vicinity of @xmath5 , and disorder induces puddles of superconducting and antiferromagnetic regions at short length - scales ; thus the system becomes an electronic micro - emulsion .
we propose that frustrated josephson couplings between the superconducting grains suppress @xmath0 .
in addition , the presence of ` normal ' quasiparticles at the interface of the frustrated josephson junctions will give rise to a highly non - trivial feature in the low - frequency response in a narrow vicinity around @xmath6 .
we propose a number of experiments to test our theory . _
introduction.- _ an important focus of the study of high temperature superconductivity ( sc ) has been on the role of antiferromagnetism ( afm ) and its relation to sc @xcite .
there is clear evidence across many different families of compounds that sc appears in close proximity to an afm phase @xcite ; these families include the iron - pnictides , the electron - doped cuprates and the heavy - fermion superconductors .
moreover , the optimal transition temperature ( @xmath7 ) of the sc is often situated where the normal state afm quantum critical point ( qcp ) would have been located , in the absence of superconductivity .
the experimental detection of the qcp is often challenging in the normal state , and more so in the superconducting state .
recently , a number of measurements were reported in a member of the pnictide family , bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of the isovalent p - doping , @xmath8 .
the experiments show a phase transition involving onset of spin - density wave ( sdw ) order in the normal state above @xmath7 , which extrapolates to a @xmath9 sdw qcp ( see @xcite and references therein ) .
these experiments include : ( _ i _ ) a sharp enhancement in the effective mass , @xmath10 , upon approaching a critical doping from the overdoped side , as obtained from de haas - van alphen oscillations @xcite and from the jump in the specific - heat at @xmath7 @xcite , and , ( _ ii _ ) a vanishing curie - weiss temperature ( @xmath11 ) , extracted from the @xmath12 measurements using nmr .
as we will review below , a number of puzzling results have appeared from experiments investigating whether the sdw qcp actually survives `` under the sc dome . '' here we propose a resolution of these puzzles by postulating a weakly first - order transition for the onset of sdw order in the presence of sc order ( see fig .
[ ph]a ) .
our results are independent of the specific microsopic mechanism responsible for rendering the transition weakly first - order @xcite .
it is well known that ` random bond ' disorder has a strong effect on symmetry - breaking first - order transitions @xcite , and ultimately replaces them with a disorder - induced second order transition in two dimensional systems .
our main claim is that the inhomogeneities associated with these highly relevant effects of disorder can resolve the experimental puzzles .
the possiblity of a qcp within the sc state was investigated by measurements @xcite of the zero temperature london penetration depth , @xmath13 ( @xmath14 superfluid - density ) , as a function of @xmath8 .
a sharp peak in @xmath15 was observed at @xmath16 and interpreted as evidence for a qcp @xcite .
however , this interpretation is at odds with general theoretical considerations @xcite concerning a qcp associated with the onset of sdw order in the presence of a superconductor with gapped quasiparticle excitations @xcite .
these considerations suggest that such systems will display a _ monotonic _ variation in @xmath15 across the qcp , rather than a sharp peak ( see dashed - blue / solid - red curves in fig .
[ ph]b ) @xcite . as a first step toward resolving this discrepancy ,
it is useful to place measurements of @xmath0 in the context of what is known about the normal state conductivity of the bafe@xmath1(as@xmath2p@xmath3)@xmath1 system , as these quantities are intimately related through a sum rule .
the low temperature superfluid density of a spatially homogeneous superconductor can be estimated from the missing area " relation , _ s_0 ^ 2/ ( z)dz , [ homese ] where @xmath17 is the elastic scattering rate and @xmath18 . in the dirty limit where @xmath19 , the above relation yields homes law @xcite , @xmath20 , whereas in the clean limit @xmath21 where @xmath22 is the conductivity spectral weight in the normal state
. eqn .
[ homese ] is particularly useful when the normal state resistivity data can reasonably be extrapolated to @xmath9 . by combining dc transport data as a function of @xmath8 @xcite and a measurement of 2@xmath23 from optical conductivity @xcite , eq .
[ homese ] provides a lower bound on @xmath15 ( with the assumption that @xmath23 is independent of @xmath8 ) .
fig.[homes ] shows @xmath15 as a function of @xmath8 obtained under this assumption ( details of the procedure are presented as supplementary information ) .
the decrease of superfluid density on the underdoped side reflects the growth in residual resistivity that begins as @xmath8 drops below about 0.33 .
the values of @xmath15 estimated from eq .
[ homese ] form a baseline for comparison with the experimental results presented in ref .
@xcite . on the same graph in fig .
[ homes ] , we show the experimentally measured @xmath15 @xcite . the data generally reflect the trend expected from the variation in the residual resistivity , with the exception of the sample with @xmath24 , in which the condensate spectral weight is suppressed by about 40% from the homes law estimate . given the constraints imposed by the sum rule ,
there are two possible sources of this discrepancy : ( _ i _ ) the quasiparticle mass could be renormalized at this value of @xmath8 , corresponding to an intrinsic decrease in @xmath22 , or , ( _ ii _ ) a considerable fraction of the ( unrenormalized ) @xmath22 could fail to contribute to the low temperature superfluid density .
the latter possibility is suggested within the scenario that we develop here .
we analyze the above experiments by assuming a weakly first - order transition @xcite , and argue that the presence of quenched disorder leads to formation of a _ micro - emulsion _ at small scales @xcite .
the system consists of sc puddles , where some of the puddles additionally have sdw order ( see fig .
[ ph]a inset ) .
the sdw(+sc ) regions , which have a locally well - developed antiferromagnetic moment but no long - range orientational order , act as barriers between the different sc grains . upon moving deeper into the ordered side of the transition , the sdw(+sc )
regions start to percolate and crossover to a state with long - range sdw order ; this is the regime with a microscopically coexistent sc+sdw . as a function of decreasing @xmath8 ,
the micro - emulsion is therefore a transitional state ( shown as grey region in fig . [ ph]a ) between a pure sc and a coexistent sc+sdw .
recent experiments in the vicinity of optimal doping using neutron - scattering and nmr have found results broadly consistent with our proposed phase diagram @xcite .
we note that the granular nature of superconductivity should have no effect on the bulk @xmath7 in the presence of percolating sc channels . _
model.- _ when the system is well described in the vicinity of @xmath5 by a micro - emulsion as explained above , the phase fluctuations associated with the sc grains ( shown as purple regions in fig .
[ ph]a inset ) , can be modeled by the following effective theory , h_= - _ a , bj_ab(_a-_b ) , where @xmath25 represent the josephson junction ( jj ) couplings between grains ` @xmath26 ' and ` @xmath27 ' .
we have ignored the capacitive contributions .
the josephson current across the junction will be given by @xmath28 , and @xmath25 may therefore be interpreted as the lattice version of the local superfluid density , @xmath29 , i.e. @xmath30 , with @xmath31 representing the superfluid - current and velocity respectively . having a frustrated jj ( also known as a @xmath32junction ) with a negative value of @xmath25 leads to a local suppression in @xmath0 .
similar ideas have been discussed in the past in a variety of contexts ( see refs .
@xcite for a specific example ) , though the mechanism considered here will be different .
we shall now propose an explicit scenario under which a suppression in @xmath0 arises in the vicinity of putative magnetic qcps , utilizing the sc gap structure in the material under question .
the basic idea is as follows : suppose that the tunneling of electrons between the two grains is mediated by the sdw moment in the intervening region @xcite , and is accompanied by a transfer of finite momentum that scatters them from a hole - like to an electron - like pocket . because the sc gaps on the two pockets have a relative phase - difference of @xmath33 , the jj coupling will be frustrated @xcite .
let us first focus on a single grain .
in order to capture the multi - band nature of the scs , we introduce two superconducting order parameters , @xmath34 with @xmath35 to model the @xmath36 state on the two pockets .
microscopically , these belong to regions in the grain having different momenta , @xmath37 , parallel to the junction .
the gaps are related to the microscopic degrees of freedom @xcite via the following relation , _
i(z)=___i v__,____- _ , where @xmath38 creates an electron at position @xmath39 with momentum @xmath37 parallel to the junction and spin @xmath40 .
@xmath41 is the pairing interaction in the cooper channel and @xmath39 is the coordinate perpendicular to the junction with area @xmath42 .
the regions @xmath43 are defined as , @xmath44 and @xmath45 , where @xmath46 is an arbitrary momentum scale chosen such that @xmath47 ( see fig . [ jj ] for an illustration ) .
we ll assume that such a prescription is valid for each grain , with possibly different values of @xmath46 .
let us then write down a model for the two coupled sc grains with an intervening proximity coupled sdw that has a well developed moment , @xmath48 .
our notation is as follows : we use @xmath49 to denote the grain index and @xmath35 to denote the band index within each grain . from now on , we relabel @xmath37 as @xmath50 .
we introduce the nambu spinor , @xmath51 , where now @xmath52 creates an electron with momentum @xmath50 parallel to the junction and at a position @xmath39 ( label suppressed ) , which belongs to a region of band @xmath53 " within grain @xmath54 " .
the effective hamiltonian is given by , [ heff ] h_&=&h_+h_t , + h_&=&_,i , _
i,,^_i,,^ , + h_t&=&g_k ( ^a_+,,[_^0]_-,,^b + & & + ^a_-,,[_^0]_+,,^b ) + , where @xmath55 is the tunneling matrix element , @xmath56 @xmath57 act in nambu space and @xmath58 @xmath57 act in spin space . in the above
, @xmath59 corresponds to the bare pairing hamiltonian written for the @xmath60 bands within each of the two grains .
@xmath61 represents the sdw moment mediated hopping of electrons from one grain to the other ( represented by the @xmath62 superscripts ) and simultaneously scattering from one band to the other ( represented by the @xmath60 subscripts ) .
therefore , @xmath48 imparts a finite momentum ( along the interface ) to the electrons when it scatters them from the electron ( hole ) pocket on one grain to the hole ( electron ) pocket on the other grain ( shown as the black arrows in fig .
[ jj ] ) . _
results.- _ using the ambegaokar - baratoff relation @xcite , we can write the josephson coupling ( at @xmath9 ) between the two grains as , j_ab= where @xmath63 and @xmath64 represent the band indices on the different grains . since @xmath65 , the coupling @xmath66 .
note that the specific nature of the frustrated tunneling arises from the same spin - fluctuation mediated mechanism that is predominantly responsible for the @xmath67 pairing symmetry @xcite .
however , there will also be a direct tunneling term ( not included in eqn .
[ heff ] ) in the hamiltonian , which does not scatter the electrons from one pocket to the other , as they hop across the junction .
the contribution to the jj coupling from this term will be unfrustrated ( i.e. @xmath68 ) .
the ratio of the tunneling amplitudes in the two different channels is non - universal and depends on various microscopic details .
in particular , the emulsion is associated with a distribution of josephson - couplings , @xmath69 , with a mean coupling strength , @xmath70 .
if a substantial fraction of the jj couplings become negative due to the mechanism proposed above , @xmath71 will be small , and the superfluid density will be suppressed ( see green curve in fig.[ph]b ) .
we now propose a resolution as to the fate of the uncondensed spectral weight ( highlighted in fig .
[ homes ] ) , which can potentially be tested by measurements of the low frequency optical conductivity .
frustrated @xmath32junctions host gapless states at the interface between the two grains @xcite , giving rise to a finite density of states around zero energy ( see fig [ sigw ] inset ) . as a result of the gapless ` normal'-fluid component at the interface , a fraction @xmath72 of the spectral weight will be displaced from the superfluid - density to non - zero frequencies ( shaded region in fig . [ sigw ] ) .
given that the weight of the condensate is proportional to @xmath73 , the 40% suppression in @xmath0 for bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of the putative qcp corresponds to @xmath74 .
our proposed optical conductivity , @xmath75 , in the vicinity of penetration depth anomaly is shown in in fig [ sigw ] .
the spectrum shows clearly that the connection between normal state conductivity and superfluid density implied by eq .
[ homes ] will break down .
in particular , @xmath76 ( which is a property of the normal state ) , could vary monotonically with isovalent - doping across @xmath6 , while the abundance of low - energy excitations in the immediate vicinity of @xmath6 would give rise to a non - monotonic variation in the superfluid density .
this allows for an unusual way of rearranging spectral weight in the _ superconducting _ state below the gap , without violating optical sum - rules
. the above scenario will give rise to a number of interesting low temperature thermodynamic and transport properties , as we now discuss .
first of all , there should be a striking enhancement in the low - temperature thermal conductivity and specific - heat , as a function of @xmath8 in the narrow vicinity of @xmath6 , due to the ` normal'-component .
it is important to recall that this material has loop - like nodes on the electron - pockets @xcite .
however , the geometry of the electron - pockets and the magnitude of the gap do not change substantially in the vicinity of @xmath6 , and therefore it is unlikely that the contribution to the above quantities from the nodal - quasiparticles will have a drastic modificiation .
it should therefore be relatively straightforward to disentangle the contribution arising from the nodal versus the ` normal ' quasiparticles .
studying the nmr - spectra as a function of decreasing temperature ( across @xmath7 ) and down to sufficiently low temperatures in the vicinity of @xmath6 should also reveal the spatial inhomogeneity associated with the sdw regions .
a large residual density of states in the superconducting state has been detected at a particular p - doping via the power - law temperature dependence of @xmath77 @xcite . within our scenario
, there should be a striking enhancement in this quantity as a function of doping around @xmath6 .
finally , we note that a promising direction for future studies would be to measure the magnetic - field distribution due to the propagating currents in the emulsion using nv - based magnetometers @xcite . _
discussion.- _ the theoretical study in this paper was motivated by a number of remarkable experiments carried out in bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of @xmath8 in the normal and superconducting phases .
our primary objective was to provide an explanation for the striking enhancement of the london penetration depth in the vicinity of a putative sdw qcp in the sc state .
we developed a scenario based on the idea that true sdw criticality is masked by a weak first - order phase transition in the superconducting state at @xmath9 . in this picture ,
quenched disorder naturally gives rise to an _ emulsion _ at small length scales with puddles of sc and sdw(+sc ) .
it is then , in principle , possible for sdw moments at the interface of the sc grains to generate frustrated josephson couplings , which deplete the local superfluid - density .
our proposed scenario naturally calls for a number of experimental tests that should be carried out in the near future , which should directly look for both the spatial inhomogeneities associated with the emulsion @xcite , and probe the gapless excitations using thermodynamic probes , as explained above .
in addition to experiments on bafe@xmath1(as@xmath2p@xmath3)@xmath1 , it should be important to further investigate the contrasting behavior of the electron - doped system , ba(fe@xmath2co@xmath3)@xmath1as@xmath1 , where @xmath78 behaves monotonically as a function of @xmath8 across the putative qcp @xcite .
electron - doping leads to significantly higher amounts of disorder compared to the isovalently - doped case , and would therefore lead to puddles with typically much smaller size @xcite .
our proposed mechanism for the strong suppression of the superfluid - density in the isovalently - doped material relies on the existence of an emulsion with puddles of appreciable size , in the presence of an optimal amount of disorder .
a comparison of the nmr spectra in the narrow vicinity of the putative qcp in the electron and isovalently doped materials would shed light on these microscopic differences between the two families . finally , though we have hypothesized that the sdw onset transition _ inside _ the sc is , in the absence of disorder , a weak first order transition , we emphasize that the normal state properties are consistent with the presence of a hidden " qcp around optimal doping @xcite .
it is plausible that in the normal state , different experimental techniques are probing the critical fluctuations associated with not one , but distinct qcps as a function of @xmath8 . for instance
, @xmath10 extracted from high - field quantum oscillations is dominated by the vicinity of ` hot - spots ' , where quasiparticles are strongly damped due to coupling to the sdw fluctuations @xcite . on the other hand , strong critical fluctuations associated with the nematic order - parameter @xcite , that couple to the entire fermi - surface ,
would dominate @xmath10 extracted at zero - field from the jump in the specific heat at @xmath7 . _
acknowledgements.- _ we thank a. carrington , a. chubukov , n. curro , j.c .
davis , r. fernandes , k. ishida , m .- h .
julien , s. kivelson , y. matsuda , a. millis and a. vishwanath for useful discussions .
we thank k. hashimoto and y. matsuda for providing us with the data shown in fig.[homes ] .
dc is supported by the harvard - gsas merit fellowship and acknowledges the boulder summer school for condensed matter physics - modern aspects of superconductivity " , where some preliminary ideas for this work were formulated .
dc and ss were supported by nsf under grant dmr-1360789 , the templeton foundation , and muri grant w911nf-14 - 1 - 0003 from aro .
ts was supported by department of energy desc-8739- er46872 , and partially by a simons investigator award from the simons foundation .
jo acknowledges the office of basic energy sciences , materials sciences and engineering division , of the u.s .
department of energy under contract no .
de - ac02 - 05ch11231 for support .
part of this work was completed when jo was visiting mit as a moore visitor supported by grant gbmf4303 .
research at perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research and innovation .
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we study the coupling of magneto - acoustic waves to alvn waves using 2.5d numerical simulations . in our experiment
, a fast magnetoacoustic wave of a given frequency and wavenumber is generated below the surface .
the magnetic field in the domain is assumed homogeneous and inclined .
the efficiency of the conversion to alfvn waves near the layer of equal acoustic and alfven speeds is measured calculating their energy flux .
the particular amplitude and phase relations between the oscillations of magnetic field and velocity help us to demonstrate that the waves produced after the transformation and reaching upper atmosphere are indeed alfvn waves .
we find that the conversion from fast magneto - acoustic waves to alfvn waves is particularly important for the inclination @xmath0 and azimuth @xmath1 angles of the magnetic field between 55 and 65 degrees , with the maximum shifted to larger inclinations for lower frequency waves .
the maximum alfvn flux transmitted to the upper atmosphere is about 23 times lower than the corresponding acoustic flux .
conversion from fast - mode high-@xmath2 magneto - acoustic waves ( analog of @xmath3 modes ) to slow - mode waves in solar active regions is relatively well studied both from analytical theories and numerical simulations ( e.g. , @xcite ) , see @xcite for a review . in a two - dimensional situation ,
the transformation from fast to slow magnetoacoustic modes is demonstrated to be particularly strong for a narrow range of the magnetic field inclinations around 2030 degrees to the vertical .
however , no generalized picture exists so far for conversion from magneto - acoustic to alfvn waves in a three - dimensional situation .
studies of this conversion were initiated by cally & goossens @xcite , who found that the conversion is most efficient for preferred magnetic field inclinations between 30 and 40 degrees , and azimuth angles between 60 and 80 degrees , and that alfvnic fluxes transmitted to the upper atmosphere can exceed acoustic fluxes in some cases .
newington & cally @xcite studied the conversion properties of low - frequency gravity waves , showing that large magnetic field inclinations can help transmitting an important amount of the alfvnic energy flux to the upper atmosphere .
time - height variations of the three projected velocity components corresponding to @xmath4 ( alfven wave , left ) , @xmath5 ( fast wave , middle ) and @xmath6 ( slow wave , right ) for @xmath7 mhz in a simulation with @xmath8 inclined by @xmath9 and @xmath10 .
the solid line marks the position @xmath11 , and the dashed line marks the cut - off layer @xmath12 .
the colour scaling is the same in all panels .
the amplitudes are scaled with @xmath13 ( first two panels ) @xmath14 ( last panel).,width=566 ] motivated by these recent studies , here we attack the problem by means of 2.5d numerical simulations .
the purpose of our study is to calculate the efficiency of the conversion from fast - mode high-@xmath2 magneto - acoustic waves to alfvn and slow waves in the upper atmosphere for various frequencies and wavenumbers as a function of the field orientation .
we limit our study to a plane parallel atmosphere permeated by a constant inclined magnetic field , to perform a meaningful comparison with the work of cally & goossens @xcite
. numerical simulation will allow generalization to more realistic models in our future work .
we numerically solve the non - linear equations of ideal mhd assuming all vectors in three spatial directions and all derivatives in two directions ( i.e. 2.5d approximation , see @xcite ) , though perturbations are kept small to approximate the linear regime .
an acoustic wave of a given frequency and wave number is generated at @xmath15 mm below the solar surface in a standard model atmosphere permeated by a uniform inclined magnetic field .
the top boundary of the simulation box is 1 mm above the surface , and 0.8 mm above the layer where the acoustic speed , @xmath16 , and the alfvn speed , @xmath17 , are equal .
we consider frequencies @xmath18 and 5 mhz and wave numbers @xmath19 mm@xmath20 and @xmath21 .
the simulation grid covers field inclinations @xmath0 from 0@xmath22 to 80@xmath22 and field azimuths @xmath1 from 0@xmath22 to 160@xmath22 .
the field strength is kept at @xmath23 g. to separate the alfvn mode from the fast and slow magneto - acoustic modes in the magnetically dominated atmosphere we use velocity projections onto three characteristic directions : @xmath24 ; \nonumber\\ { \hat\mathbf{e}}_{\rm perp } & = & [ - \cos\phi \sin^2\theta \sin\phi , \ , 1-\sin^2\theta \sin^2\phi , \ , - \cos\theta \sin\theta \sin\phi ] ; \\
\nonumber { \hat\mathbf{e}}_{\rm trans } & = & [ -\cos\theta , \ , 0 , \ , \cos\phi \sin\theta].\end{aligned}\ ] ] to measure the efficiency of conversion to alfvn waves near and above the @xmath11 equipartition layer , we calculate acoustic and magnetic energy fluxes , averaged over time : @xmath25 figure [ fig : modes ] shows an example of the projected velocities in our calculations as a function of space and time . in this representation
the larger inclination of the ridges mean lower propagation speeds and vice versa . note , that by projecting the velocities , we are able to separate the modes only in the magnetically dominated atmosphere , i.e. above the solid line in fig . [
fig : modes ] .
the figure shows how the incident fast mode wave propagates to the equipartition layer and then splits into several components .
the alfvn wave is produced by mode conversion above 0.2 mm ( left panel ) and propagates upwards with the ( rapid ) alfvn speed , confirmed by almost vertical inclination of the ridges .
conversely , the essentially magnetic fast - mode low-@xmath2 wave produced in the upper atmosphere ( middle panel ) is reflected , and its velocity variations in the upper layers vanish with height . the ( acoustic ) slow - mode low-@xmath2 wave escapes to the upper atmosphere tunnelling over the cut - off layer due to the field inclination of @xmath9 .
the amplitudes of the velocity variations of the alfvn wave are comparable to those of the slow wave .
left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 .
black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) .
right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 .
red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 .
black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) .
right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 .
red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] to confirm the alfvn nature of the transformed waves , as revealed by the projection calculations , we checked the amplitude and phase relations for all three modes reaching the upper atmosphere . for the alfvn mode the magnetic field @xmath27 and velocity variations
@xmath30 should be in equipartition ( i.e. @xmath31 ) , and both magnitudes should oscillate in phase ( see priest @xcite ) .
figure [ fig : phases ] presents the calculations of the amplitude ratio @xmath32 and temporal phase shift between @xmath27 and @xmath30 , where both velocity and magnetic field variations are projected in the corresponding characteristic direction for each mode ( eq . [ eq : directions ] ) .
this calculation confirms that , indeed , for all magnetic field orientations @xmath0 and @xmath1 , the amplitude ratio for the alfvn mode ( @xmath29 projection ) is around one ( left panel ) .
this is clearly not the case for the slow and fast modes . for the fast mode ,
the amplitude ratio is two orders of magnitude larger , and for the slow mode , it is two orders of magnitude lower than one . for the alfvn mode the phase shifts group around zero for all @xmath1 , unlike the case of the fast mode ( right panel ) .
we did not calculate the phase shifts for the slow mode as the variations of the magnetic field are negligible .
thus , we conclude that the properties of the simulated alfvn mode separated by the projection correspond to those expected for a classical alfvn mode . examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq .
[ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] an example of the height variations of the acoustic and magnetic fluxes is given in figure [ fig : fluxes2 ] .
the total vertical flux ( dotted line ) is conserved in the simulations except for the limitations caused by the finite grid resolution not resolving slow small - wavelength waves in the deep layers ( see fig . [
fig : modes ] ) .
both acoustic and magnetic fluxes show strongest variations near the conversion layer and become constant above it between 0.5 and 1 mm height .
the fluxes reaching the upper atmosphere depend crucially on the orientation of the field . in this example
, the acoustic flux decreases with @xmath0 whilst the magnetic flux increases with @xmath0 and becomes larger than the acoustic fluxes for @xmath33 . as the fast wave
is already reflected in the upper atmosphere ( see fig . [
fig : modes ] ) , the magnetic flux at these heights is due to the propagating alfvn wave .
vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) .
upper panels give magnetic fluxes and lower panels give acoustic fluxes .
, title="fig:",width=226 ] vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) .
upper panels give magnetic fluxes and lower panels give acoustic fluxes .
, title="fig:",width=226 ] finally , figure [ fig : fluxes ] gives the time averages of the vertical magnetic and acoustic fluxes at the top of the atmosphere as a function of the field orientation .
as proven above , the magnetic flux at 1 mm corresponds to the alfvn mode . at @xmath7 mhz
, the maximum of the magnetic flux corresponds to @xmath34 and @xmath35 .
this maximum is shifted to larger inclinations @xmath36 for waves with @xmath18 mhz .
the presence of the sharp maximum of the alfvnic flux transmission agrees well with the conclusions made previously by cally & goossens @xcite , though the exact position of the maximum is shifted to somewhat larger inclinations .
the maximum of the transmitted acoustic flux corresponds to inclinations @xmath37 for @xmath7 mhz waves , and to @xmath38 for @xmath18 mhz waves , again , in agreement with previous calculations @xcite .
the absolute value of the fluxes is about 30 times lower for 3 mhz compared to 5 mhz . at some angles the afvn magnetic flux transmitted to the upper atmosphere is larger than the acoustic flux .
however , at angles corresponding to the maximum of the transmission , the alfvn flux is 2 - 3 times lower than the corresponding acoustic flux .
it is important to realize that quantitatively simulating mode transformation numerically is a challenge , as any numerical inaccuracies are amplified in such second - order quantities as wave energy fluxes .
the tests presented in this paper prove the robustness of our numerical procedure and offer an effective way to separate the alfvn from magneto - acoustic modes in numerical simulations .
this will allow us in future to study the coupling between magneto - acoustic and alfvn waves in more realistic situations resembling complex solar magnetic structures .
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we would like to thank f.s .
navarra for fruitiful conversations .
this work has been partly supported by fapesp and cnpq - brazil . for a review and references to original works ,
see e.g. , s. narison , _ qcd as a theory of hadrons , cambridge monogr . part .
* 17 * , 1 ( 2002 ) [ hep - h/0205006 ] ; _ qcd spectral sum rules , world sci .
notes phys . _
* 26 * , 1 ( 1989 ) ; acta phys . pol . *
b26 * , 687 ( 1995 ) ; riv .
* 10n2 * , 1 ( 1987 ) ; phys . rept . *
84 * , 263 ( 1982 ) .
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we use the qcd sum rules to evaluate the mass of a possible scalar mesonic state that couples to a molecular @xmath0 current .
we find a mass @xmath1 gev , which is in a excellent agreement with the recently observed @xmath2 charmonium state .
we consider the contributions of condensates up to dimension eight , we work at leading order in @xmath3 and we keep terms which are linear in the strange quark mass @xmath4 . we also consider a molecular @xmath5 current and we obtain @xmath6 , around 200 mev above the mass of the @xmath7 charmonium state .
we conclude that it is possible to describe the @xmath2 structure as a @xmath8 molecular state .
there is growing evidence that at least some of the new charmonium states recently discovery in the b - factories are non conventional @xmath9 states . some possible interpretations for these states
are mesonic molecules , tetraquarks , or / and hybrid mesons .
some of these new mesons have their masses very close to the meson - meson threshold like the @xmath10 @xcite and the @xmath11 @xcite .
therefore , a molecular interpretation for these states seems natural .
the most recent aquisiton for this list of peculiar states is the narrow structure observed by the cdf collaboration in the decay @xmath12 .
the mass and width of this structure is @xmath13 , @xmath14 @xcite .
since the @xmath2 decays into two @xmath15 vector mesons , it has positive @xmath16 and @xmath17 parities .
there are already some theoretical interpretations for this structure .
its interpretation as a conventional @xmath9 state is complicated because , as pointed out by the cdf collaboration @xcite , it lies well above the threshold for open charm decays and , therefore , a @xmath9 state with this mass would decay predominantly into an open charm pair with a large total width . in ref .
@xcite , the authors interpreted the @xmath2 as the molecular partner of the charmonium - like state @xmath7 , which was observed by belle and babar collaborations near the @xmath18 threshold @xcite .
they concluded that the @xmath2 is probably a @xmath19 molecular state with @xmath20 or @xmath21 . in ref .
@xcite they have interpreted the @xmath2 as an exotic hybrid charmonium with @xmath22 . in this work ,
we use the qcd sum rules ( qcdsr ) @xcite , to study the two - point function based on a @xmath8 current with @xmath20 , to see if the new observed resonance structure , @xmath2 , can be interpreted as such molecular state . in previous calculations ,
the hidden charm mesons @xmath23 and @xmath24 have been studied using the qcdsr approach as tetraquark or molecular states @xcite . in some cases a very good agreement with the experimental mass was obtained . the starting point for constructing a qcd sum rule to evaluate the mass of a hadronic state , @xmath25 ,
is the correlator function ( q)=id^4x e^iq.x0 |t[j_h(x)j_h^(0)]|0 , where the current @xmath26 creates the states with the quantum numbers of the hadron @xmath25 .
a possible current describing a @xmath27 molecular state with @xmath28 is j=(|s_a_c_a)(|c_b^s_b ) , [ field ] where @xmath29 and @xmath30 are color indices .
the qcd sum rule is obtained by evaluating the correlation function in eq .
( [ 2po ] ) in two ways : in the ope side , we calculate the correlation function at the quark level in terms of quark and gluon fields .
we work at leading order in @xmath3 in the operators , we consider the contributions from condensates up to dimension eight and we keep terms which are linear in the strange quark mass @xmath4 . in the phenomenological side , the correlation function is calculated by inserting intermediate states for the @xmath27 molecular scalar state . parametrizing the coupling of the scalar state , @xmath31 , to the current , @xmath32 , in eq .
( [ field ] ) in terms of the parameter @xmath33 : [ eq : decay ] 0 | j|h=. [ lam ] the phenomenological side of eq .
( [ 2po ] ) can be written as ^phen(q^2)=^2m_h^2-q^2+_0^ds ^cont(s)s - q^2 , where the second term in the rhs of eq.([phe ] ) denotes higher scalar resonance contributions .
it is important to notice that there is no one to one correspondence between the current and the state , since the current in eq .
( [ field ] ) can be rewritten in terms of sum a over tetraquark type currents , by the use of the fierz transformation .
however , the parameter @xmath33 , appearing in eq . ( [ lam ] ) , gives a measure of the strength of the coupling between the current and the state .
the correlation function in the ope side can be written as a dispersion relation : ^ope(q^2)=_4m_c^2^ds
^ope(s)s - q^2 , where @xmath34 is given by the imaginary part of the correlation function : @xmath35 $ ] . as usual in the qcd sum rules method
, it is assumed that the continuum contribution to the spectral density , @xmath36 in eq .
( [ phe ] ) , vanishes bellow a certain continuum threshold @xmath37 . above this threshold , it is given by the result obtained with the ope .
therefore , one uses the ansatz @xcite ^cont(s)=^ope(s)(s - s_0 ) , to improve the matching between the two sides of the sum rule , we perfom a borel transform . after transferring the continuum contribution to the ope side , the sum rules for the scalar meson , considered as a scalar @xmath38 molecule , up to dimension - eight condensates , using factorization hypothesis , can be written as : ^2e^-^2/m^2=_4m_c^2^s_0ds e^-s / m^2 ^ope(s ) , [ sr1 ] where ^ope(s)=^pert(s)+^(s ) + ^g^2(s)+^mix(s)+^^2(s)+^mix(s ) , with [ eq : pert ] & & ^pert(s)=32 ^ 9 ^6_^ d^3 _ ^1-d^2(1 - - ) ^3(-4m_cm_s ) , + & & ^(s)=32 ^ 5 ^ 4_^ d\{m_s(m_c^2-(1-)s)^21 - - m_c_^1-d .
+ & & . }
, + & & ^g^2(s)=m_c^22 ^ 8 ^ 6_^ d^3_^1-d(1 - - ) , + & & ^mix(s)=-m_0 ^ 22 ^ 6 ^ 4\ { 3m_c_^d [ m_c^2-(1-)s ] -m_s(8m_c^2-s ) } , + & & ^^2(s)=m_c^28 ^ 2\ { ( 2m_c - m_s)-m_sm_c^2_0 ^ 1d ( s - m_c^2(1- ) ) } , [ dim6 ] where the integration limits are given by @xmath39 , @xmath40 , @xmath41 , and we have used @xmath42 .
we have neglected the contribution of the dimension - six condensate @xmath43 , since it is assumed to be suppressed by the loop factor @xmath44 .
we also include a part of the dimension-8 condensate contributions , related with the mixed condensate - quark condensate contribution : ^mix(s)&=&-m_cm_0 ^ 2 ^ 216 ^ 2_0 ^ 1 d ( s - m_c^2(1- ) ) .
[ dim8 ] it is important to point out that a complete evaluation of the dimension-8 condensate , and higher dimension condensates contributions , require more involved analysis @xcite , which is beyond the scope of this calculation . to extract the mass @xmath45 we take the derivative of eq .
( [ sr ] ) with respect to @xmath46 , and divide the result by eq .
( [ sr ] ) . for a consistent comparison with the results obtained for the other molecular states using the qcdsr approach , we have considered here the same values used for the quark masses and condensates as in refs .
@xcite : @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 with @xmath52 , @xmath53 .
the borel window is determined by analysing the ope convergence and the pole contribution . to determine the minimum value of the borel mass we impose that the contribution of the dimension-8 condensate should be smaller than 20% of the total contribution . in fig .
[ figconv ] we show the contribution of all the terms in the ope side of the sum rule . from this figure
we see that for @xmath54 gev@xmath55 the contribution of the dimension-8 condensate is less than 20% of the total contribution .
therefore , we fix the lower value of @xmath56 in the sum rule window as @xmath57 gev@xmath55 . the maximum value of the borel mass is determined by imposing that the pole contribution must be bigger than the continuum contribution . in table
i we show the values of @xmath58 . in our numerical analysis , we will consider the range of @xmath56 values from 2.3 @xmath59 until the one allowed by the pole dominance criterion given in table i. + [ cols="^,^",options="header " , ] taking into account the incertainties given above we finally arrive at = ( 4.140.09 ) , [ ymass ] in an excellent agreement with the mass of the narrow structure @xmath2 observed by cdf .
one can also deduce , from eq .
( [ sr1 ] ) , the parameter @xmath33 defined in eq .
( [ lam ] ) .
we get : = ( 4.220.83 ) 10 ^ -2 ^5 , [ la1 ] from the above study it is very easy to get results for the @xmath60 molecular state with @xmath20 . for this
we only have to take @xmath61 and @xmath62 in eqs .
( [ dim6 ] ) , ( [ dim8 ] ) .
this study was already done in ref .
@xcite considering @xmath63 .
although in the case of the @xmath64 scalar molecule we get a worse borel convergence than for the @xmath38 scalar molecule , as can be seen by fig .
[ opedd ] , there is still a good ope convergence for @xmath65 . if we allow also for the @xmath60 molecule values of the continuum threshold in the range @xmath66 we get @xmath67 .
therefore , from a qcd sum rule study , the difference between the masses of the states that couple with scalar @xmath8 and @xmath60 currents , is consistent with zero . the mass obtained with the @xmath60 scalar current is about 100 mev above the @xmath68 threshold .
this could be an indication that there is a repulsive interaction between the two @xmath69 mesons .
strong interactions effects might lead to repulsive interactions that could result in a virtual state above the threshold .
therefore , this structure may or may not indicate a resonance .
however , considering the errors , it is not compatible with the observed @xmath70 charmonium - like state . in fig .
[ dif ] we show the relative ratio @xmath71 as a function of the borel mass for @xmath72 .
from this figure we can see that the ratio is very stable as a function of @xmath56 and the difference between the masses is smaller than 0.5% .
although the ratio is shown for @xmath72 , the result is indiscernible from the one shown in fig .
[ dif ] for other values of the continuum threshold in the range @xmath73 .
this result for the mass difference is completely unexpected since , in general , each strange quark adds approximately 100 mev to the mass of the particle .
therefore , one would naively expect that the mass of the @xmath38 state should be around 200 mev heavier than the mass of the @xmath64 state .
this was , for instance , the result obtained in ref .
@xcite for the vector molecular states @xmath74 and @xmath75 , where the masses obtained were : @xmath76 and @xmath77 . for the value of the parameter @xmath33
we get : _ d^*d^ * = ( 4.200.96)10 ^ -2 ^5 . [ la2 ] therefore , comparing the results in eqs .
( [ la1 ] ) and ( [ la2 ] ) we conclude that the currents couple with similar strength to the corresponding states , and that both , @xmath8 and @xmath60 scalar molecular states have masses compatible with the recently observed @xmath2 narrow structure .
however , since the @xmath2 was observed in the decay @xmath78 , the @xmath8 assignment is more compatible with its quark content . in conclusion , we have presented a qcdsr analysis of the two - point function for possible @xmath8 and @xmath60 molecular states with @xmath20 .
our findings indicate that the @xmath2 narrow structure observed by the cdf collaboration in the decay @xmath12 can be very well described by using a scalar @xmath8 current . although the authors of ref .
@xcite interpreted the @xmath2 as a @xmath27 molecular scalar state and the @xmath7 as a @xmath5 molecular scalar state , we have obtained similar masses for the states that couple with the scalars @xmath27 and @xmath5 currents .
therefore , from a qcd sum rule point of view , the charmonium - like state @xmath7 , observed by belle and babar collaborations , has a mass around 200 mev smaller than the state that couples with a @xmath5 scalar current and , therefore , can not be well described by such a current .
while this work has been finalized , a similar calculation was presented in ref .
@xcite .
however , the author of ref .
@xcite arrived to a different conclusion .
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we thank g. kohring , d. stauffer and c. tsallis for interesting discussions .
this work was performed within the sfb 341 kln
aachen jlich supported by the dfg .
[ fig2 ] time - dependence of the activity for @xmath42 and various values of @xmath13 .
the system size is @xmath72 . concerning the statistical error we observe that all runs using different random numbers yield curves that are indistinguishable on this scale .
[ fig3 ] determination of @xmath43 : the system size is @xmath73 , @xmath42 and @xmath68 ( lower curve ) , @xmath57 ( middle curve ) and @xmath74 ( upper curve ) .
the straight line in the middle is the function @xmath75 .
we conclude that @xmath76 .
[ fig5 ] time - dependence of the distance for @xmath77 and @xmath78 ( lower curve ) , @xmath69 ( upper curve ) .
the system size is @xmath79 .
the curve for @xmath69 shows that @xmath80 is well inside the chaotic region , whereas @xmath81 lies within the frozen phase .
this fact stronly supports reentrance .
the emerging phase diagram in the vicinity of the tricritical point is depticted in the insert the two black dots represent the parameter values for the two curves shown .
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we present numerical and analytical results for a special kind of one - dimensional probabilistic cellular automaton , the so called domany - kinzel automaton .
it is shown that the phase boundary separating the active and the recently found chaotic phase exhibits reentrant behavior .
furthermore exact results for the @xmath0=0-line are discussed .
pacs numbers : 87.10.+e , 02.50.+s , 89.80.+h cellular automata have been an intensive research field in recent years @xcite due to their computational simplicity and the wide range of applications in various areas . even in one dimension
a particular probabilistic variant ( domany - kinzel automaton ) of the originally deterministic cellular automata shows a rich phase diagram including directed percolation and other critical phenomena @xcite . only recently
a new phase in this model has been explored numerically exhibiting chaotic behavior @xcite .
this region of the diagram , up to a deterministic corner - point , is not accessible to exact treatments up to now .
nevertheless sophisticated approximation - methods , which systematically go beyond mean - field theory , have been applied successfully @xcite . in the so called tree - aproximation @xcite one
finds reentrant behavior in two directions , which is not fully understood yet .
this phenomenon has never been observed in numerical simulations up to now @xcite .
therefore one might ask , whether this reentrant behavior is a real feature of the model or just an artifact of the tree - approximation .
this issue is the main topic of the present paper , where we try to clearify this point with an alternative approximation method ( the cluster - approximation ) as well as with large scale monte - carlo simulations ( up to @xmath1 sites ) . to state the final results already at this place : the cluster - approximation again yields reentrant behavior in two directions and the simulations show clear evidence for reentrance near the tricritical point .
the model we consider is defined as follows : the domany - kinzel pca consists of a one - dimensional chain of @xmath2 binary variables , @xmath3 , @xmath4 taking on the values @xmath5 ( empty , occupied ) .
all sites are updated simultaneously ( i.e.parallel ) at discrete time steps and the state of each site at time @xmath6 depends only upon the state of the two nearest neighbors at time @xmath7 according to the following rule : @xmath8 \right\ } \end{aligned}\ ] ] where @xmath9 is the ( time - independent ) conditional probability that site @xmath10 takes on the value @xmath4 given that its neighbors have the values @xmath11 and @xmath12 at the previous time step .
@xmath13 ( @xmath0 ) is the probability that site @xmath10 is occupied if exactly one ( both ) of its neighbors is ( are ) occupied .
if neither neighbor is occupied , the site @xmath10 will also become empty , therefore the state with all sites empty is the absorbing state of the pca . the @xmath14-phase diagram , as it is known up to now , consists of three different phases .
most of it ( small enough @xmath13 ) is dominated by the _ frozen _
phase , where all initial conditions eventually lead into the absorbing state .
with other words , the activity @xmath15 tends to zero for @xmath16 within the frozen phase . for large enough @xmath13
one enters the @xmath17 phase , where , starting from a random initial condition , the system ends up in a state with a finite density of active sites . within this active phase
one can distinguish between a chaotic and a non - chaotic part .
this difference can be seen by starting with two slighly different ( random ) initial conditions @xmath18 and
@xmath19 subjected to the same external noise ( local updating rules ) . calculating the normalized distance @xmath20 of these two systems @xmath21 during the update of the replicated systems according to the rule displayed in equation ( 2 ) of reference @xcite one
observes a sharp transition from the chaotic phase , characterized by @xmath22 , to the active phase with @xmath23 ( in the following we call the active / non - chaotic phase simply the active phase ) .
the underlying picture is that in the latter case the system is characterized by only one attractor , which nevertheless depends strongly on the external noise . with other words , in this phase the noise ( and not the initial condition ) dominates the dynamics completely .
this is not true for the chaotic phase , where the system memorizes the initial state even after infinite time .
first we present analytical results obtained by the application of the so - called cluster - approximation already known in different contexts @xcite as probability path method @xcite or local structure theory @xcite . in this way we check earlier results @xcite derived with a different approximation scheme ( the tree - approximation ,
see @xcite ) .
the problem with the dynamical rules defined above is that one can not write down the probability distribution of the stationary state since no simple detailed balance condition can be derived .
therefore , in principle , it is necessary to solve the dynamics completely in order to obtain the equilibrium properties .
this is not possible in general .
one way out of this dilemma is to take into account systematically all possible correlations between @xmath24 neighboring sites ( @xmath24-cluster approximation ) and to treat interactions over longer distances by conditional probabilities .
more formally , given the probability @xmath25 for the configuration @xmath26 in an @xmath24-cluster - approximation the probabilitiy for configuration @xmath27 with @xmath28 is approximated to be : @xmath29 here @xmath30 denotes the conditional probability to find site @xmath31 in state @xmath32 given that the @xmath33 sites to the left are in the state @xmath34 .
a factorisation of this kind can describe the stationary state exactly only if the interactions extend over not more than @xmath24 sites .
a natural choice for the conditional probaility @xmath35 is @xmath36 with @xmath37 simple examples of one - dimensional systems which can be described exactly by a finite value of @xmath24 are the @xmath38-spin - ising - model where one needs @xmath39 for the exact equilibrium distribution ( @xmath40 being the standard one - dimensional ising model with next - neighbour interactions only ) @xcite .
another example is the the parallel asymmetric exclusion process where again @xmath41 leads to the exact result for the stationary state @xcite the phase diagram resulting from a calculation based on the cluster approximation with @xmath41 is shown in figure 1 .
since during one update step according to the rules equation 1 the even ( odd ) sites only depend on the odd ( even ) sites at the timestep before we performed two timesteps at once to deal with sites of only one fixed parity .
one firstly observes that @xmath41 is still far from the exact solution for the stationary state .
unfortunately higher approximations are very hard to obtain due to the exponentially growing number of equations to be analysed simultaneously ( especially for the distance @xmath20 with two replicated systems ) .
furthermore even for @xmath41 the resulting equations can not be solved analytically with final closed expressions but have to be iterated until one finds a fixed point of the system of equations . in order to obtain a better localisation of the phase boundaries we applied the same method described below to analyse the numerical data from the monte - carlo - simulations . as can be seen from the figure we find reentrant behaviour both in @xmath13- and @xmath0-direction comparable to the result from the tree approximation @xcite .
it seems that the tricritical point has moved upwards , but a detailed analysis of the results suggests that it remains on the @xmath42-line . for the frozen / active - phase boundary one can go to larger clusters with higher values of @xmath24 . in tabular 1 the critical values @xmath43 ( @xmath42 ) for of @xmath44
are given : @xmath45 a simple least square fit leads to a limiting value for @xmath43 of about @xmath46 which is significantly larger than the known values from the simulations @xcite . in order to test the predictions of both approximation schemes mentioned above we performed large - scale monte - carlo simulations of the domany - kinzel cellular automaton with probabilities @xmath13 and @xmath0 in the vicinity of the two end - points of the phase boundary of the chaotic phase ( i.e. : @xmath47 and @xmath48 ) , where reentrance could occur according to the above calculations .
the system - sizes were up to @xmath49 sites with periodic boundary conditions , and the number of iterations @xmath50 were maximally @xmath51 . in this way
one avoids self - correlations ( finite size effects ) , since after @xmath7 updates those sites separated by a distance smaller than @xmath7 are correlated .
therefore @xmath50 has to be smaller than @xmath2 . by choosing @xmath2 much larger than @xmath50 one improves the statistics significantly ( for obvious reasons , since one can devide the system into many statistically independent subsystems ) .
therefore no finite - size effects are present in our data ( which was checked by comparing results for different system sizes ) and we need not to perform a ( non - trivial ) extrapolation the infinite system @xmath52 .
furthermore the probability that the system gets trapped by the absorbing state ( @xmath53 ) after time @xmath7 increases with decreasing system size .
this renders the simultaneous limit @xmath52 and @xmath54 to a delicate point , which we also avoid by our approach .
looking at the data obtained from the simulations it turned out to be rather unreliable to try to discriminate between two phases by looking at the long - time limit of the order parameter ( activity @xmath55 or distance @xmath20 ) . apart from the two phase - boundaries we expect exponential decay of @xmath55 and @xmath20 to their asymptotic values .
exactly on the phase - boundary we expect the spectrum of relaxation times to extend to infinity and thus the decay to become algebraic .
this behavior is illustrated in figure 2 : the activity as a function of time is depicted in a log - log plot for increasing values of @xmath13 ( @xmath42 ) .
we see that below a certain value the curves are bended downwards , whereas above this value the curves are bended upwards reflecting exactly the behaviour explained above .
the curve just in the middle corresponding to @xmath56 is closest ( as defined quantitatively by a least square fit ) to a straight line . to determine @xmath43 more accurately we performed longer runs with larger system sizes and depict the result in figure 3 . the middle curve , corresponding to @xmath57 , is nicely approximated by an algebraic decay with an exponent @xmath58 .
this exponent agrees well with the universal order parameter exponent @xmath59 determined in reference @xcite . from figure 3
we determined @xmath43 to be @xmath60 .
this is the most accurate estimate of @xmath43 so far .
surprinsingly it is significantly larger than the value @xmath61 obtained with a different method but with system sizes of around @xmath62 @xcite .
it seems that in the latter reference the long transient times ( @xmath63 ) together with the small system sizes lower the critical value due to larger correlations in the system as known from similar systems @xcite . as we have mentioned above the finite size scaling analysis of small systems
is by no means straightforward and can not be done without further ad hoc assumptions , from which our method is free .
hence , from our point of view , the results we quote seem to be more reliable .
note that there is no overlap even of the error bars of the two critical values . in figure 4
we show the same scenario for @xmath64 . by the same arguments as above we now locate the critical value of @xmath13 ( i.e.the value at which the transition from vanishing to finite activity takes place ) to be @xmath65 , which is significantly lower than @xmath43 . for larger increasing values of @xmath0
the phase boundary between frozen and active phase bends down monotonically to smaller values of @xmath13 terminating at the point ( @xmath66 , @xmath67 ) which is exactly known since the whole @xmath67-line is exactly solvable . in figure 5
a comparison of the two curves for @xmath20 at @xmath68 , which is below @xmath43 , for @xmath42 and @xmath69 is shown .
note that ( 0.8090,0 ) lies within the frozen phase .
the upper curve bends upwards , which means that ( 0.8090,0.03 ) lies within the chaotic phase .
this is indicated by the schematic phase diagram depicted in the insert of figure 5 and which has been supported by simulations of various parameters @xmath14 in this region .
the two black dots represent the two curves shown and along the arrow connecting them one finds clear evidence for reentrant behavior .
the phase boundary of the chaotic phase therefore bends to the left up to values around @xmath69 and for larger values of @xmath0 it bends monotonically to the right until it terminates at the point @xmath48 .
one reason for the fact that this phenomenon was not seen in earlier simulations is that it is in fact a marginal effect observable only in high - precision simulation - data .
we also performed large scale simulations around the other endpoint of the chaotic / active phase boundary . here
it is quite evident that the reentrant behaviour parallel to the @xmath13-axis at @xmath48 is in fact an artifact of the approximation schemes and not existent in the actual system . concerning the question of the conjugate field for the order - parameter of the chaotic phase posed in reference @xcite one can make on the @xmath0=0-line exact statements .
since both the activity and the chaos order parameter obey exactly the same evolution equations @xcite it is easy to conclude that the conjugated fields also should be equivalent . for the activity one chooses independent random numbers at each site and each timestep ( on the @xmath0=0-line
this is just the role of @xmath13 ) .
accordingly one chooses for the chaotic order parameter independant random numbers at each site and each timestep ( rule @xmath70 in ref.@xcite ) .
the absorbing state now corresponds to identical variables states in the two replicas yielding the same update since the same noise has to be applied for equal configurations in the two systems .
this picture remains valid for @xmath71 although the evolution equations are no longer identical , but the absorbing state has the same properties .
note that on the line @xmath42 the critical exponents of the order - paramater are also the same .
if universality holds away from this line this statement should be true also for @xmath71 ( for @xmath67 it is known , that the critical exponents are different @xcite ) . in summary
we have shown in this letter that , in contradiction to previous findings , the chaotic phase in fact shows reentrant behaviour in the vicinity of the tricritical point as predicted by approximative analytical methods .
the effect was not seen before since it is relatively small and large scale simulations have to be made to detect it . on the other hand in the near of the @xmath13=1-line the predicted reentrant behaviour
is absent .
furthermore one can see that simulations of small systems with long transient times can lead to erroneous conclusions about the locations of the critical point as well as the shape of the phase boundary since it is difficult to estimate the error due to self correlations .
therefore the error - bars in reference @xcite seem to neglect these systematic errors and should be larger ( which could lead to an agreement with our results ) .
finally we have seen that the conjugate field to the chaotic order parameter can directly be identified from the equivalence to the activity order parameter on the @xmath0=0-line .
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we would like to thank bruce reed for introducing us to the classical version of the locker puzzle and richard cleve for pointing out the perfect quantum search of @xcite .
this work was partially supported by an an nserc discovery grant and an nserc postdoctoral fellowship .
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the _ locker puzzle _ is a game played by multiple players against a referee .
it has been previously shown that the best strategy that exists can not succeed with probability greater than , no matter how many players are involved .
our contribution is to show that quantum players can do much better they can succeed with . by making the rules of the game significantly stricter ,
we show a scenario where the quantum players still succeed perfectly , while the classical players win with vanishing probability .
other variants of the locker puzzle are considered , as well as a cheating referee .
* keywords : quantum complexity , grover search , locker puzzle * 10000 10000 grover s quantum algorithm @xcite provides a quadratic speedup over the best possible classical algorithm for the problem of unsorted searching in the query model . while grover s search method has been shown to be optimal @xcite , our results reveal that in the context of multi - player query games , applying grover s algorithm yields success probabilities that are much better than the success probabilities of classical optimal protocols .
specifically , we show that in the case of the _ locker puzzle _ , quantum players succeed with probability 1 while the known optimal classical success probability is bounded above by . in order to amplify this separation , we prove that a significantly stricter version of the locker puzzle has vanishing classical success probability , while still admitting a perfect quantum strategy .
we also consider the empty locker and the coloured slips versions of the locker puzzle , and the possibility of a cheating referee .
[ sec : locker puzzle ] the _ _ locker puzzle _ _ is a cooperative game between a team of @xmath0 players numbered @xmath1 and a referee . in the initial phase of the game , the referee chooses a random permutation @xmath2 of @xmath3 , and for each player @xmath4 she places number @xmath4 in locker @xmath5 . in the following phase , each player
is individually admitted into the locker room .
once in the room , each player is allowed to open @xmath6 lockers , one at a time , and look at their contents ( for simplicity , we ll take @xmath0 to be even ) .
after the player leaves the room , all lockers are closed .
the players are initially allowed to discuss strategy , but once the game starts , they are separated and can not communicate .
an individual player @xmath4 _ wins _ if he opens a locker containing number @xmath4 , while the team of @xmath0 players _ wins _ if all individual players win .
we would like to know what is the best strategy for the team of @xmath0 players .
a nave approach is for each player to independently choose @xmath6 lockers to open .
each players wins independently with probability @xmath7 , hence the team wins with probability @xmath8 .
surprisingly , it is known that the players can do much , much better !
we will review in section [ section : optimal - classical - locker ] an optimum strategy by which , for any @xmath0 , the players can win with probability at least @xmath9 .
the locker puzzle was originally considered by peter bro miltersen , and was first published in @xcite ; a journal version appears in @xcite .
sven skylum is credited for the pointer - following strategy that we will give in the next section .
a proof of optimality for this strategy is given by eugene curtin and max warshauer @xcite .
our presentation of the classical puzzle and its solution follows along the lines of their article .
many variations have been proposed @xcite .
we will consider the variations of _ empty lockers _ in section [ section : empty - lockers ] , _ coloured slips _ in section [ section : coloured - slips ] ( to be accurate , the locker and the coloured slips puzzles are variants of the empty locker puzzle ) , and a _ cheating referee _ in section [ sec : cheating ] .
[ section : optimal - classical - locker ] we saw that a nave solution allows the players to win with an exponentially small probability .
how can we devise a strategy that does better ? the reader avid to search for a solution on his or her own is encouraged to do so now .
the key is to find a solution where the individual success probabilities are not independent .
consider the following strategy : when first entering the locker room , player @xmath4 opens locker number @xmath4 .
a number is revealed ; this is used to indicate which locker to open next ( i.e. if number @xmath10 is revealed , the next locker opened is locker @xmath10 ) .
each player executes this pointer - following strategy until @xmath6 lockers are opened . to analyze the success probability , note that the team will win provided that the _ permutation _ that corresponds to the placement of numbers in lockers by the referee does not contain a cycle of length longer than @xmath6 .
the probability of such a long cycle occurring is : @xmath11 it can be shown that as @xmath12 , @xmath13 and that the sum increases with @xmath0 .
hence the probability that the team wins is decreasing to @xmath14 . using a reduction to another game
, this strategy can be shown to be optimal @xcite .
[ sec : quantum solution ] we now present our first contribution : a quantum solution to the locker puzzle , which performs better than the classical solution . as before the referee chooses a random permutation @xmath2 and she places numbers in the lockers according to this permutation . in the quantum solution , we allow the players to open locker doors in _ superposition _ , each player working with his own quantum register .
this is analogous to the quantum query model . for the quantum case
, we need to modify the goal of the game which , for player @xmath4 , becomes to _ correctly guess _ the locker containing number @xmath4 after @xmath6 queries , and _ not _ to open locker containing number @xmath4 , because this would be too easy to do in superposition ! we show that quantum players can always win at the locker game .
in fact , our results are stronger : we give a stricter version of the locker puzzle for which the optimal classical solution succeeds with vanishing probability , while a quantum strategy always succeeds ! [ imp ]
the main idea is to apply grover s quantum search algorithm to the locker puzzle . for player @xmath4
, we consider the action of opening a locker as a query to the oracle which when input locker number @xmath15 , @xmath16 , outputs the following : @xmath17 note that this oracle is weaker than the oracle in the original puzzle which would output @xmath18 .
we discuss this further in section [ subsection : optimality ] and in the conclusion .
grover s search algorithm @xcite was thoroughly analyzed in @xcite , where it was shown that in a black - box search scenario where it is known that a single solution exists , @xmath19 queries yield a failure probability no greater than @xmath20 , where @xmath0 is the number of elements in the search space ( here , @xmath0 is assumed to be large ) .
this was further improved in @xcite , where is was shown that the same amount of queries is sufficient to find a solution with _
certainty_.
applying this directly to the quantum players of the locker puzzle yields the following : 1 .
[ step : groverquery]each player performs @xmath19 queries ( this is less than the @xmath21 queries in the classical solution ) .
2 . each player wins independently with certainty , implying that the team wins with certainty .
[ sec : reducing - number - queries ] we ve seen that quantum players of the locker game can succeed with probability 1 .
our solution only requires @xmath19 oracle queries per player .
hence , we now consider the asymptotically stricter version of the locker puzzle , where players are allowed to open at most @xmath22 lockers .
the next theorem state that the success probability for classical players goes quickly to 0 .
[ thm : classic ] in the locker puzzle with @xmath22 queries , classical players win with probability at most @xmath23 .
let @xmath24 .
we upper bound the success probability of the first @xmath25 players , when each player is allowed to open @xmath26 lockers . since @xmath27 ,
this upper bounds the success probability of all @xmath0 players .
consider a new game where the first player opens exactly @xmath26 lockers and publicly reveals all of their contents . if the first player s number is not revealed the players lose and the game is over .
otherwise the @xmath26 revealed players have successfully located their lockers .
these @xmath26 lockers and players are now removed from the game .
the first player has success probability at most @xmath28 . in successive rounds ,
a player is chosen from amongst those not yet removed from the game .
he continues in the same way by choosing @xmath26 of the remaining lockers and revealing their contents .
if he finds his label , again @xmath26 lockers and players are removed from the game .
the game stops whenever a chosen player does not find his label .
otherwise it continues for @xmath26 rounds and terminates with a win for the players .
the success probability of the new game is at most @xmath29 the original game with no revealing of numbers can not do better . [
subsection : optimality ] in the quantum query model with oracle ( [ eq : oracle ] ) the total number of queries required to obtain a success probability of one for the players is in @xmath30 .
first consider a variation of the quantum game where the players act sequentially in the order @xmath31 and are allowed to announce their results to the other players .
the number of queries performed by player 1 must be in @xmath32 or he will not succeed with probability one .
this follows from the analysis of grover s algorithm , see @xcite . the only information given by the oracle @xmath33 is the location of the locker containing label @xmath34 .
suppose player 2 is allowed to receive this information and remove that locker from consideration .
the permutation @xmath2 induces a random permutation on the remaining @xmath35 lockers .
player 2 s success probability is then one only if his number of queries is in @xmath36 .
continuing , the @xmath4-th player must ask a number of queries in @xmath37 .
the total number of queries is therefore in @xmath30 . in the modified game we share all information available to all players that have not already played .
so this shows a lower bound of the same order for the original version of the quantum game where no information is shared .
let us now compare the strength of oracle ( [ eq : oracle ] ) with the stronger oracle where @xmath38 . in the classical setup
, the weaker oracle ( [ eq : oracle ] ) merely tells a given player whether or not his label is in a requested locker .
there are an even number @xmath0 of lockers and he can ask @xmath39 queries . again
we consider a sequential version of the game as described above , where each player reveals his results .
if he succeeds , he reveals the locker with his number and that locker is removed . for the other lockers he queried
, the only information he has is that they did not contain his label .
therefore after his locker is removed , the other players have no further information .
the success probability of this variation of the locker game is : @xmath40 where we have used stirling s formula twice .
this is exponentially small and provides an upper bound on the success probability of the classical locker game with the weak oracle ( [ eq : oracle ] ) . by comparison , as we saw in section [ section : optimal - classical - locker ] the players can win with constant probability using the stronger oracle .
an open question is whether the quantum algorithm can be improved by using this stronger oracle .
the original motivation for the locker puzzle came from the study of time - space tradeoffs for the substring search problem in the context of _ bit probe complexity _
@xcite . there , a version with both _
empty lockers _ and _ coloured slips _ was presented .
we now examine these two variations separately and consider the quantum case .
[ section : empty - lockers ] suppose there are a total of @xmath41 lockers .
the referee selects an unordered subset @xmath2 of @xmath42 with cardinality @xmath0 and she puts label @xmath4 into locker @xmath43 for @xmath44 .
the remaining @xmath45 lockers are empty .
assume @xmath46 is even , and we allow the players to open up to @xmath47 lockers .
an optimum winning strategy for this more general situation is unknown : the pointer algorithm fails if an empty locker is opened .
even for the case @xmath48 , where half of the lockers are empty , it is still unknown if there is a classical strategy with success probability bounded away from zero @xcite .
however , the quantum strategy given in section [ sec : quantum solution ] still succeeds with probability one with a number of queries in @xmath49 per player , for a total of @xmath50 queries .
it suffices to modify the oracle ( [ eq : oracle ] ) so that @xmath15 runs over the range @xmath51 , and query it @xmath52 times .
if it turns out that for these same parameters , the classical success probability vanishes , then the power of the quantum world would be once more confirmed , as in section [ sec : reducing - number - queries ] . and section [ subsection : optimality ] .
[ section : coloured - slips ] consider the empty lockers game with @xmath41 lockers , again with @xmath0 players and @xmath0 slips of paper , each labelled @xmath53 .
this time the referee colours each slip either red or blue as she chooses , and places them in a randomly selected subset of @xmath0 lockers . as before ,
each player @xmath4 may open @xmath47 lockers using any adaptive strategy , and based on this , must make a guess about the colour of the slip labelled @xmath4 .
the players win if every player correctly announces the colour of his slip . with @xmath54 ,
this can be solved with the pointer - following algorithm and the players have success probability about 0.31 . in the quantum setting
, the players can win with probability one at the colour guessing game also , by changing the oracle ( [ eq : oracle ] ) .
let @xmath55 be the colour of the slip for player @xmath4 .
define for @xmath51 and @xmath56 : @xmath57 now we use the protocol described in section [ imp ] with each player querying this new oracle @xmath52 times .
if for player @xmath4 @xmath58 , then there is exactly one @xmath15 for which @xmath59 and grover s algorithm returns @xmath60 with probability one .
otherwise , if @xmath61 then @xmath62 is identically zero and grover s algorithm may return any value @xmath15 .
the player now makes one further call to oracle ( [ eq : oracle1 ] ) with the returned value @xmath15 and guesses red if the oracle returns one and blue otherwise .
[ sec : cheating ] a cheating referee can obviously beat the players in the locker game .
she simply has to omit the label of one of the players .
this could be easily exposed by requiring that all the lockers be opened and checked at the end of the game .
a more subtle way of cheating is if the referee can somehow choose the permutation @xmath2 . in the original locker game ,
let @xmath63 , and let @xmath64 be a random unordered subset of @xmath65 players .
she may set @xmath66 , @xmath67 , and fill out the rest of @xmath2 at random from the remaining players .
it is easy to verify that , using the pointer algorithm , player @xmath68 opens @xmath6 lockers @xmath69 and does not find his label .
he has to guess and loses with probability about @xmath70 .
the same thing happens for each of the players @xmath71 .
( incidentally , the reason for not choosing @xmath72 is that the players not finding their label may guess the locker number they see in the last locker they open , winning the game with probability one ! ) .
using variants of this idea the referee may cheat successfully for some time before the players catch on .
if the players have access to shared randomness ( which is unknown to the referee ) , they can circumvent this problem by first applying their own permutation on the lockers before opening any of them .
interestingly , our quantum protocol is impervious to a referee who maliciously chooses the permutation , and does not require shared randomness .
[ sec : conclusion ] it was previously known that the locker puzzle has an intriguing classical optimal solution .
now we know that the locker puzzle and its variants also have interesting quantum solutions which perform significantly better than the classical ones . we have given a quantum solution in the black - box query complexity model that _ does not use the pointer - following technique that is crucial to the classical optimal solution_. it would be interesting to see if using the stronger classical oracle could lead to a quantum solution that works with a reasonable probability of success using @xmath73 total queries . with this stronger oracle ,
perhaps shared entanglement could help the players ?
it would also be interesting to see if , analogous to the classical case , our results have any consequences for time - space tradeoffs for data structures @xcite .
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