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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm166, 40 pages, doi:10.1093/imrn/rnm166 published on February 11, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Birth of a Cut in Unitary Random Matrix Ensembles

Tom Claeys

Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Department of Mathematical Sciences, Brunel University West London, Uxbridge UB8 3PH UK

Correspondence: Correspondence to be sent to: tom.claeys{at}wis.kuleuven.be

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.


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