The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the "birth of a cut" transition

TL;DR

This paper analyzes the double scaling limit of eigenvalues in a random matrix ensemble near a critical transition point, using Riemann-Hilbert methods to derive asymptotics and connect to known kernels.

Contribution

It introduces a rigorous Riemann-Hilbert approach to describe the eigenvalue behavior near the 'birth of a cut' transition, extending previous results by Eynard.

Findings

Asymptotic correlation kernel near the critical point is identified.

The kernel matches that of a random unitary ensemble with weight e^{-x^{2 u}}.

Provides a rigorous proof of earlier heuristic results.

Abstract

In this paper we studied the double scaling limit of a random unitary matrix ensemble near a singular point where a new cut is emerging from the support of the equilibrium measure. We obtained the asymptotic of the correlation kernel by using the Riemann-Hilbert approach. We have shown that the kernel near the critical point is given by the correlation kernel of a random unitary matrix ensemble with weight $e^{-x^{2\nu}}$. This provides a rigorous proof of the previous results of Eynard.