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Painlevé Representation of Tracy–Widom\({_\beta}\) Distribution for \({\beta}\) = 6

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Abstract

In Rumanov (J Math Phys 56:013508, 2015), 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 ordinary differential equation (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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  1. Department of Applied Mathematics, CU Boulder, Boulder, CO, USA

    Igor Rumanov

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  1. Igor Rumanov
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Correspondence to Igor Rumanov.

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Communicated by P. Deift

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Rumanov, I. Painlevé Representation of Tracy–Widom\({_\beta}\) Distribution for \({\beta}\) = 6. Commun. Math. Phys. 342, 843–868 (2016). https://doi.org/10.1007/s00220-015-2487-5

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