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Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

Title:

Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

Bertola, Marco, Buckingham, R., Lee, S. Y. and Pierce, V. (2012) Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes. Journal of Statistical Physics, 146 (3). pp. 475-518. ISSN 0022-4715

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Official URL: http://dx.doi.org/10.1007/s10955-011-0409-2

Abstract

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. 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. For a Gaussian potential, it was shown by Péché (Probab. Theory Relat. Fields 134:127–173, 2006) that when r is fixed or grows sufficiently slowly with n (a small-rank source), r eigenvalues are expected to exit the main bulk for |a| large enough. Furthermore, at the critical value of a when the outliers are at the edge of a band, the eigenvalues at the edge are described by the r-Airy kernel. We establish the universality of the r-Airy kernel for a general class of analytic potentials for r=O(n γ ) for 0≤γ<1/12.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Bertola, Marco and Buckingham, R. and Lee, S. Y. and Pierce, V.
Journal or Publication:Journal of Statistical Physics
Date:2012
Digital Object Identifier (DOI):10.1007/s10955-011-0409-2
Keywords:Riemann-Hilbert problem Asymptotic analysis Nonlinear steepest descent analysis r-Airy kernel Critical phenomena
ID Code:976934
Deposited By: DANIELLE DENNIE
Deposited On:05 Mar 2013 15:58
Last Modified:18 Jan 2018 17:43

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