Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The heat flow conjecture for random matrices   Recent results by various authors have established a "model deformation
phenomenon" in random matrix theory. Specifically, it is possible to construct
pairs of random matrix models such that the limiting eigenvalue distributions
are connected by push-forward under an explicitly constructible map of the
plane to itself. In this paper, we argue that the analogous transformation at
the finite-$N$ level can be accomplished by applying an appropriate heat flow
to the characteristic polynomial of the first model.
  Let the "second moment" of a random polynomial $p$ denote the expectation
value of the square of the absolute value of $p.$ We find certain pairs of
random matrix models and we apply a certain heat-type operator to the
characteristic polynomial $p_{1}$ of the first model, giving a new polynomial
$q.$ We prove that the second moment of $q$ is equal to the second moment of
the characteristic polynomial $p_{2}$ of the second model. This result leads to
several conjectures of the following sort: when $N$ is large, the zeros of $q$
have the same bulk distribution as the zeros of $p_{2},$ namely the eigenvalues
of the second random matrix model. At a more refined level, we conjecture that,
as the characteristic polynomial of the first model evolves under the
appropriate heat flow, its zeros will evolve close to the characteristic curves
of a certain PDE. All conjectures are formulated in "additive" and
"multiplicative" forms.
  As a special case, suppose we apply the standard heat operator for time $1/N$
to the characteristic polynomial $p$ of an $N\times N$ GUE matrix, giving a new
polynomial $q.$ We conjecture that the zeros of $q$ will be asymptotically
uniformly distributed over the unit disk. That is, the heat operator converts
the distribution of zeros from semicircular to circular.

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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: cs