Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continuous and discrete Painleve equations arising from the gap
  probability distribution of the finite $n$ Gaussian Unitary Ensembles   In this paper we study the gap probability problem in the Gaussian Unitary
Ensembles of $n$ by $n$ matrices : The probability that the interval $J :=
(-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van
Moerbeke and Forrester and Witte on this subject, it has been shown that two
Painleve type differential equations arise in this context. The first is the
Jimbo-Miwa-Okomoto $\sigma-$form and the second is a particular Painleve IV.
Using the ladder operator technique of orthogonal polynomials we derive three
quantities associated with the gap probability, denoted by $\sigma_n(a)$,
$R_n(a)$ and $r_n(a)$, and show that each one satisfying a second order,
non-linear, differential equation as well as a second order, non-linear
difference equation. In particular, in addition to providing an elementary
derivation of the aforementioned $\sigma-$form and Painleve IV we show that the
quantity $r_n(a)$ satisfies a particular case of Chazy's second degree second
order differential equation. For the discrete equations we show that the
quantity $r_n(a)$ satisfies a particular form of the modified discrete Painleve
II equation obtained by Grammaticos and Ramani in the context of Backlund
transformations. We also derive second order second degree difference equations
for the quantities $R_n(a)$ and $\sigma_n(a)$.

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The categories of this paper are: alg-geom
1
The categories of this paper are: cs
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The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat