Large deviations of the extreme eigenvalues of random deformations of matrices

• 成果类型: Article
• 署名作者: Benaych-Georges, F.; Guionnet, A.; Maida, M.
• 署名单位: Sorbonne Universite; Universite Paris Cite; Ecole Normale Superieure de Lyon (ENS de LYON); Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0382-3
• 发表日期: 2012
• 页码: 703-751
• 关键词: spectral measure; Moderate Deviations; UNIVERSALITY

摘要

Consider a real diagonal deterministic matrix X (n) of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of X (n) converge to the edges of the support of the limiting measure and when we allow some eigenvalues of X (n) , that we call outliers, to converge out of the bulk. We can also generalise our results to the case when X (n) is random, with law proportional to e (-n Tr V(X)) dX, for V growing fast enough at infinity and any perturbation of finite rank.