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

### Abstract

We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \mathrm{Tr} V(M)}dM,$ 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 $q”=sq+2q^3-\alpha$. 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.

## Authors

Tom Claeys

Departement Wiskunde
Katholieke Universiteit Leuven
3001 Leuven
Belgium

Arno B. J. Kuijlaars

Departement Wiskunde
Katholieke Universiteit Leuven
3001 Leuven
Belgium

Maarten Vanlessen

Departement Wiskunde
Katholieke Universiteit Leuven
3001 Leuven
Belgium