Painlevé representation of Tracy-Widom\(_\beta\) distribution for \(\beta=6\)

DOI10.1007/S00220-015-2487-5zbMATH Open1383.60010arXiv1408.3779OpenAlexW3126099480MaRDI QIDQ261618

Author name not available (Why is that?)

Publication date: 24 March 2016

Published in: (Search for Journal in Brave)

Abstract: In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $$. Using this general result, the case $$ is further considered here. This is the smallest even $$, when the corresponding Lax pair and its relation to Painlev'e II (PII) have not been known before, unlike cases $$ 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 $$ 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 $-2/3$.

Full work available at URL: https://arxiv.org/abs/1408.3779

Mathematics Subject Classification ID

No records found.

This page was built for publication: Painlevé representation of Tracy-Widom\(_\beta\) distribution for \(\beta=6\)

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q261618)