Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gap Probabilities for Double Intervals in Hermitian Random Matrix
  Ensembles as $\tau$-Functions -- Spectrum Singularity case   The probability for the exclusion of eigenvalues from an interval $(-x,x)$
symmetrical about the origin for a scaled ensemble of Hermitian random
matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $
a $ (a generalisation of the sine kernel in the bulk scaling case), is
considered. It is shown that this probability is the square of a
$\tau$-function, in the sense of Okamoto, for the Painlev\'e system \PIII. This
then leads to a factorisation of the probability as the product of two
$\tau$-functions for the Painlev\'e system \PIIIdash. A previous study has
given a formula of this type but involving \PIIIdash systems with different
parameters consequently implying an identity between products of
$\tau$-functions or equivalently sums of Hamiltonians.

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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: funct-an
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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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: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn