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Birth of a Cut in Unitary Random Matrix Ensembles

Publisher: OUP

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 o...View more

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.
Published in: International Mathematics Research Notices ( Volume: 2008, Issue: 9, 2008)
Page(s): rnm166 - rnm166
Date of Publication: 2008

ISSN Information:

Publisher: OUP