The birth of a cut in unitary random matrix ensembles

Abstract

We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It turns out that limits of the eigenvalue correlation kernel are given by Hermite kernels corresponding to a finite size Gaussian Unitary Ensemble (GUE). When modifying the double scaling limit slightly, we observe a remarkable transition each time the new cut picks up an additional eigenvalue, leading to a limiting kernel interpolating between GUE-kernels for matrices of size k and size k+1. We prove our results using the Riemann-Hilbert approach.

arXiv comment28 pages, 2 figures

References in corpus (4)

• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Universality of a double scaling limit near singular edge points in random matrix models
• Universal distribution of random matrix eigenvalues near the "birth of a cut" transition
• Multi-critical unitary random matrix ensembles and the general Painleve II equation

Cited by in corpus (5)

• Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
• A new type of critical behaviour in random matrix models
• The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the "birth of a cut" transition
• Mesoscopic colonization of a spectral band
• First colonization of a hard-edge in random matrix theory