Bertola, Marco, Buckingham, R., Lee, S. Y. and Pierce, V. (2012) Spectra of Random Hermitian Matrices with a SmallRank External Source: The Critical and NearCritical Regimes. Journal of Statistical Physics, 146 (3). pp. 475518. ISSN 00224715

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Official URL: http://dx.doi.org/10.1007/s1095501104092
Abstract
Random Hermitian matrices are used to model complex systems without timereversal 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 smallrank 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 rAiry kernel. We establish the universality of the rAiry 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/s1095501104092 
Keywords:  RiemannHilbert problem Asymptotic analysis Nonlinear steepest descent analysis rAiry 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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