Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

2010/09/20 by Bertola, Marco, Buckingham, Robert, Lee, Seung-Yeop +1
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1009.3894

Abstract

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \citeAdler:2009a, Daems:2007 and sample covariance matrices \citeBaik:2005. We consider the case when the n× n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n-r. The source is small in the sense that r is finite or r=\mathcal O(n^γ), for 0< γ<1. For a Gaussian potential, Péché \citePeche:2006 showed that for |a| sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for |a| sufficiently large (the supercritical regime) r eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of r× r Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.

Related

• On the largest eigenvalue of a Hermitian random matrix model with spiked external source II. Higher rank cases
• Gaussian matrix model in an external field and non-intersecting Brownian motions
• On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case
• Universality in the random matrix spectra in the regime of weak non-Hermiticity
• Almost-Hermitian random matrices and bandlimited point processes