Universality of the double scaling limit in random matrix models

TL;DR

This paper proves the universality of eigenvalue correlation kernels in certain critical random matrix models using Riemann-Hilbert analysis and Painleve functions, extending previous results to more general potentials.

Contribution

It establishes universality at quadratic vanishing points of eigenvalue density for unitary ensembles, generalizing prior work to all cases with a novel approach allowing negative densities.

Findings

Universality of eigenvalue correlation kernels at critical points.

Construction of kernels using Painleve II functions.

Extension of previous universality results to more general potentials.

Abstract

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift/Zhou steepest descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly.