Eigenvalue Separation in Some Random Matrix Models

Abstract

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

25 pages, 9 figures included

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• Dyson's non-intersecting Brownian motions with a few outliers

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• Subsystem dynamics under random Hamiltonian evolution
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Probability densities and distributions for spiked and general variance Wishart $\beta$-ensembles
• Distribution of rare saddles in the $p$-spin energy landscape
• Scaling limits of correlations of characteristic polynomials for the Gaussian beta-ensemble with external source
• Ensemble Inequivalence in the Spherical Spin Glass Model with Nonlinear Interactions
• Spectral density of the non-central correlated Wishart ensembles
• Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean
• Onset of universality in the dynamical mixing of a pure state
• Top eigenpair statistics of diluted Wishart matrices
• Edge effects in some perturbations of the GUE
• Emergence of a spectral gap in a class of random matrices associated with split graphs