Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Painlev\'e representation of Tracy-Widom$_\beta$ distribution for $\beta
  = 6$   In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta
ensembles with even integer values of $\beta$. Using this general result, the
case $\beta=6$ is further considered here. This is the smallest even $\beta$,
when the corresponding Lax pair and its relation to Painlev\'e II (PII) have
not been known before, unlike cases $\beta=2$ and $4$. It turns out that again
everything can be expressed in terms of the Hastings-McLeod solution of PII. In
particular, a second order nonlinear ODE for the logarithmic derivative of
Tracy-Widom distribution for $\beta=6$ involving the PII function in the
coefficients, is found, which allows one to compute asymptotics for the
distribution function. The ODE is a consequence of a linear system of three
ODEs for which the local singularity analysis yields series solutions with
exponents in the set $4/3$, $1/3$ and $-2/3$.

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The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph