Multi-critical unitary random matrix ensembles and the general Painleve II equation

Abstract

We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)}dM$, where $\alpha >-1/2$ and $V$ is such that the limiting mean eigenvalue density for $n,N\to \infty$ and $n/N\to 1$ vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight $|x{|}^{2\alpha }{e}^{-NV(x)}$. Here the main focus is on the construction of a local parametrix near the origin with $\psi$-functions associated with a special solution $q_{\alpha }$ of the Painlevé II equation $q^{\prime \prime }=sq+2q^3-\alpha$. We show that $q_{\alpha }$ has no real poles for $\alpha >-1/2$, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of $q_{\alpha }$ in the double scaling limit.

37 pages, 4 figures

References in corpus (5)

• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation
• Universality in Random Matrix Theory for orthogonal and symplectic ensembles
• Universality of the double scaling limit in random matrix models
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Cited by in corpus (2)

• Lectures on random matrix models. The Riemann-Hilbert approach
• The birth of a cut in unitary random matrix ensembles