Painlevé representation of Tracy-Widom${}_{\beta }$ distribution for $\beta =6$

Abstract

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 Painlevé 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 $-2/3$.

arXiv commentpresentation improved, references added

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Cited by in corpus (7)

• Extreme value statistics of correlated random variables: a pedagogical review
• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• On the Tracy-Widom${}_{\beta }$ Distribution for $\beta =6$
• "Quantizations" of isomonodromic Hamiltonian Garnier system with two degrees of freedom
• Hamiltonian reductions in Matrix Painlevé systems
• Computing the Tracy-Widom Distribution for Arbitrary $\beta >0$