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

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.