Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: {\bf $\tau$-Function Evaluation of Gap Probabilities in Orthogonal and
  Symplectic Matrix Ensembles}   It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
$\tau$-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
$\tau$-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two $\tau$-functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
$\tau$-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those $\tau$-functions is the gap probability in
the corresponding orthogonal symmetry case.

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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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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: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an