appendix.tex
\appendix{}
\section{Appendix: Zonal spherical functions}
\label{app1}
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A remarkable feature of the theory of zonal spherical functions (ZSF)
discussed in subsection \ref{sec-zonal} is that in some special cases
exact formulae exist for these functions. This happens in particular
for all the symmetric spaces with multiplicity of ordinary roots $m_o=2$
in the classification of Table~\ref{tab1} or Table~\ref{tab3}. The
ultimate reason for this becomes clear by inspection of the mapping
discussed in section \ref{sec-intmod}. The radial part of the
Laplace--Beltrami operator can be transformed into a suitable
Hamiltonian of $n$ interacting particles which decouple exactly for
$m_o=2$.  Thus we may expect to be able to write the ZSF's in these
particular cases as a suitable determinant of single-particle
eigenfunctions.

In the simplest case in which the multiplicity of both the short and long
roots is zero one can write

\eq
\phi_\lambda \propto [\xi(q)]^{-1}\sum_{s\in W}\det s \e^{i(s\lambda,q)}
\label{eq:app1}
\en

where $W$ denotes the Weyl group (which in this case is simply the permutation group, while
$\det s$ is simply the sign
of the permutation $s$), $\lambda$ labels the ZSF and is
related to its eigenvalue (see eq. (\ref{eq:eigen})), $\xi(q)$ is
given by eq. (\ref{eq:xiJ}) and $(\lambda,q)$ denotes the vector
product.   
In the above expression we recognize, as anticipated, the determinant 
$\det e^{\lambda_iq_j}$ written as an
alternating sum over the permutation group $W\equiv S_n$.
 Notice, as a side remark, that if the curvature of the symmetric space is
positive, then $\lambda$ is ``quantized" and must belong to the dual
weight lattice.

This expression fixes the $q$--dependence of the ZSF. However, it does
not completely fix the $\lambda$--dependence, which is hidden in the
proportionality constant. Fixing this constant turns out to be a
highly non--trivial task. In the case of the $A_n$ type spaces it can
be obtained by direct integration of eq. (\ref{eq:sphericalfunctions-})
\cite{OlshPere}. The final result (in which we have introduced
explicitly the parameter $a$ as in paragraph \ref{sec-zonal} so as to
describe the three possible curvatures in one single equation) is

\eq
\phi^{(-)}_\lambda(aq)=\frac{1!\,2!\,...\,(n-1)!\sum_{s\in W}
\det s \,\e^{i(s\lambda,q)}}
{\prod_{j<k}\frac{\lambda_j-\lambda_k}{a} ~\prod_{j<k} \sinh [a(q_j-q_k)]}
\label{eq:app2}
\en
 
The extension to other root lattices is slightly more involved.  Due
to the presence of the long and/or short roots or to the additional
reflection symmetry of the lattice, the corresponding ``single
particle'' Schr\"odinger equations are more complicated and the
solutions are no longer simple plane waves. The problem was solved in
1958 by Berezin and Karpelevick~\cite{bk58}. The solution they
obtained is valid for a generic $BC_n$ lattice and, remarkably enough,
holds for {\it any} value of $m_s$ and $m_l$, not only for those
related to the symmetric spaces (while obviously $m_o=2$ is
mandatory).
 
In this case there is no compact way to deal with the three possible curvatures
in a single formula, so let us look at the three cases separately:
 
{\bf Positive curvature spaces}  

\eq
\phi_\lambda^{(+)}(q)=C_+\frac{ \det\left[\frac{\eta_l!\,\Gamma(\alpha+1)}
{\Gamma(\eta_l+\alpha+ 1)}~P^{\alpha,\beta}_{\eta_l}(\cos2q_j)\right]}
{\prod_{j<k}(\lambda_j^2-\lambda_k^2)~
\prod_{j<k}(\sin^2q_j^2-\sin^2q_k^2)}
\label{eq:app3}
\en

where the $P^{\alpha,\beta}_{\eta_l}$ are Jacobi polynomials with
$\alpha=(m_s+ m_l + 1)/2$, $\beta=(m_l-1)/2$ and
$\eta_l=(2\lambda_l-m_s-2m_l)/4$ (recall that in this case $\lambda$
is quantized, thus $\eta_l$ turns out to be an integer number). The
normalization constant $C$ does not contain any further dependence on
$\lambda$ and $q$. Its value is $C=2^{n(n-1)}
\prod_{l=1}^{n-1}l!\,(m_s+ l)^{n-l}$.

{\bf Negative curvature spaces}

\eq
\phi_\lambda^{(-)}(q)=C_-\frac{ \det\left[F(a_l,b_l,c;-\sinh^2q_j)\right]}
{\prod_{j<k}(\lambda_j^2-\lambda_k^2)~
\prod_{j<k}(\sinh^2q_j^2-\sinh^2q_k^2)}
\label{eq:app4}
\en

where this time $F(a_l,b_l,c;-\sinh^2q_j)$ denotes the hypergeometric
function and the three parameters are :
$a_l=(m_s+2m_l+2i\lambda_l)/4$, $b_l=(m_s+2m_l-2i\lambda_l)/4$ and
$c=(m_s+m_l+1)/2$.  The normalization constant is the same as in the
positive curvature case. This is the formula that we used in paragraph
\ref{sec-beta2} to solve the DMPK equation in the $\beta=2$ case.

{\bf Zero curvature spaces}  

In this case the expression is slightly simpler:

\eq
\phi_\lambda^{(0)}(q)=C_0\frac{ \det\left[(\lambda_lq_j)^{-\gamma}
J_\gamma(\lambda_lq_j)\right]}
{\prod_{j<k}(\lambda_j^2-\lambda_k^2)~
\prod_{j<k}(q_j^2-q_k^2)}
\label{eq:app5}
\en

where $\gamma={(m_s+ m_l-1)/2}$ and

\eq
C_0=\frac{(-1)^{n(n-1)/2}\,2^{n(n+\hat{m}
-3/2)}\,\Gamma(\hat{m}+1/2)\,...\,\Gamma(\hat{m}+n-1/2)}{1!\,2!\,3!\,...\,
(n-1)!}
\en
 
with $\hat{m}=(m_s+m_l)/2$.

\subsection{The Itzykson--Zuber--Harish--Chandra integral}

A remarkable feature of the zero curvature case discussed above is
that not only the zonal spherical functions themselves are simpler,
but also the integral representation from which they are obtained
drastically simplifies. In fact, in this case (recall that $m_o=2$)
the symmetric spaces coincide with the classical Lie algebras and
eq. (\ref{eq:sphericalfunctions-}) can be written as a simple integral
over the group manifold. For instance in the unitary case, i.e. for
systems of type $A_n$, the integral representation becomes
 
\eq
\label{eq:app6}
\phi_\lambda^{(0)}(q)=\int D U e^{{\rm Tr} \left( \Lambda U Q U^{\dagger}
\right)}
\en

where $U\in SU(n)$, $\Lambda$ and $Q$ are diagonal matrices:
$\Lambda={\rm diag}(\lambda_1,...,\lambda_n)$ and $Q={\rm diag}(q_1,...,q_n)$.

On the other hand, if we inspect the $a\to0$ limit of
eq. (\ref{eq:app2}) we find:

\eq
\phi^{(0)}_\lambda(q)=\frac{1!\,2!\,...\,(n-1)!\sum_{s\in W}
\det s \,\e^{i(s\lambda,q)}}
{\prod_{j<k}(\lambda_j-\lambda_k) \prod_{j<k} (q_j-q_k)}
\label{eq:app7}
\en
 
In this last expression we recognize the $\det e^{\lambda_iq_j}$ written as an
alternating sum over the symmetric group $W\equiv S_n$, while the two products
in the denominator can be interpreted as Vandermonde determinants

\eq
\Delta(\lambda) = \prod_{i<j} (\lambda_i-\lambda_j)
\label{vandermond}
\en

Moreover, we easily see that eq. (\ref{eq:app6}) holds unchanged if
$\Lambda$ and $Q$ are two generic hermitean matrices with eigenvalues
$\{\lambda_i\}$ and $\{q_i\}$, respectively. At this point, equating
these two expressions for $\phi^{(0)}$ we find the remarkable
relation:

\eq
\label{eq:app8}
\int D U e^{{\rm tr} \left( \Lambda U Q U^{\dagger} \right)}
=\frac{1!\,2!\,...\,(n-1)!\det \e^{\lambda_iq_j}} {\Delta(\lambda)\Delta(q)}
\en

which is the well--known Itzykson--Zuber--Harish--Chandra (IZHC in the
following) integral.  This result was originally obtained by
Harish--Chandra in~\cite{hc57}. It was later rediscovered in the
context of random matrix models by Itzykson and Zuber~\cite{iz80} and
fully exploited by Mehta in~\cite{m81}. It plays a major role in
several physical applications of random matrix ensembles, ranging from
the exact solution of two--dimensional QCD to the study of parametric
correlations in random matrix theories.
 
It is not difficult, using equation (\ref{eq:app5}), to generalize
eq.~(\ref{eq:app8}) to other Lie groups, i.e. to other root
lattices. This general result was already present in the original
paper by Harish--Chandra~\cite{hc57}. A recent review with detailed
expression for $SO(n)$ (both for even and odd $n$) and $Sp(n)$ can be
found for instance in appendix A of~\cite{bhl99}.


\subsection{The Duistermaat--Heckman theorem}

The remarkable elegance and simplicity of the IZHC integral suggests
that there should be some deep mathematical principle underlying this
result. Indeed, in the last few years it has been realized that the
IZHC integral is a particular example of a wide class of integrals
which may be solved {\it exactly} with the saddle point method
provided one sums over all the critical points (and not only over the
maxima). In the IZHC case this means that one has to sum over all the
elements of the Weyl group (i.e over all the permutations in the
unitary case) and not only over the ones with positive signature (as
one would do in a standard saddle point approximation). 

The rationale behind this remarkable result is the
Duistermaat--Heckman theorem \cite{dh82}. It states that the saddle
point is exact (provided one sums over all the critical points) if the
integration is performed over an orbit with a symplectic structure.
As a matter of fact, it was later recognized that this is only an
instance of a wider class of results which are known as {\it 
localization theorems}. Their common feature is that they can be used
to reduce integrals over suitably chosen manifolds to sums over sets
of critical points. A more detailed dicussion of this beautiful branch
of modern mathematics goes beyond the scopes of the present review. To
the interested reader we suggest reference \cite{bt95}, where a
comprehensive and readable review on the localization formulae and
some their physical applications can be found.




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DFTT 10/03\\
\end{flushright}
\vspace{10mm}
%
\begin{center}
{\large 
{\bf Random matrix theory and symmetric spaces
}}\\[15mm]
%
{%\sc
\bf 
M.~Caselle$^1$ and U.~Magnea$^2$ 
} \\
\vspace{5mm}
{\it $^1$Department of Theoretical Physics, University of Torino \\
and INFN, Sez. di Torino\\
Via P. Giuria 1, I-10125 Torino, Italy \\ 
$^2$Department of Mathematics, University of Torino \\ 
Via Carlo Alberto 10, I-10123 Torino, Italy}\\
caselle@to.infn.it \\
magnea@dm.unito.it
\\[6mm]
\vspace{13mm}
%
{\bf Abstract}\\[5mm]
\end{center}

In this review we discuss the relationship between random matrix
theories and symmetric spaces. We show that the integration manifolds
of random matrix theories, the eigenvalue distribution, and the Dyson and
boundary indices characterizing the ensembles are in strict
correspondence with symmetric spaces and the intrinsic characteristics
of their restricted root lattices. Several important results can be obtained
from this identification. In particular the Cartan classification of triplets of 
symmetric spaces with
positive, zero and negative curvature gives rise to a new
classification of random matrix ensembles.
The review is organized into two main parts. 
In Part~I the theory of symmetric spaces is
reviewed with particular emphasis on the ideas relevant for
appreciating the correspondence with random matrix theories.  
In Part~II we discuss various applications of symmetric spaces to random matrix
theories and in particular the new
classification of disordered systems derived from the classification
of symmetric spaces. We also review how the mapping from integrable
Calogero--Sutherland models to symmetric spaces can be used in the
theory of random matrices, with particular consequences for quantum
transport problems. We conclude indicating some interesting new directions of
research based on these identifications.

\end{titlepage}

\newpage
\setcounter{footnote}{0}

\tableofcontents
\input{sec_intro}
\newpage
\begin{center}
\Large{\bf Part I}
\end{center}
\input{sec_ss}
\input{sec_intmod}
\newpage
\begin{center}
\Large{\bf Part II}
\end{center}
\input{sec_rmt}
\input{sec_appl}
\input{sec_proposal}
\input{sec_concl}

\vskip 1cm
{\bf Acknowledgments}
This work was partially supported by the 
European Commission TMR program HPRN-CT-2002-00325 (EUCLID).
U.~M. wishes to thank Prof. S. Salamon for discussions.
\newpage
\input{appendix}

\newpage

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