Random Matrices with Merging Singularities and the Painlevé V Equation

Abstract

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n}\left|\det \right(M^2-tI){|}^{\alpha }{e}^{-n\mathrm{Tr}V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $\alpha >-1/2$ and $t\in \mathbb{R}$, in double scaling limits where $n\to \infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of $\alpha$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

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Cited by in corpus (3)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems