Abstract
We consider non-Hermitian random matrices X∈Cn×n with general decaying correlations between their entries. For large n, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear n×n matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of X. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.
Highlights
We consider non-Hermitian random matrices X ∈ Cn×n with general decaying correlations between their entries
We prove that the Brown measure of these matrixvalued circular elements has the properties (i), (ii), (iii) listed above
To resolve the instability we perform a non-linear transformation of the matrix Dyson equation (MDE) that allows to restrict the analysis to the manifold of perturbations that respect the additional symmetry of M
Summary
Many random matrix models exhibit a strong concentration of measure phenomenon; their empirical eigenvalue distributions are well approximated by deterministic measures as their sizes tend to infinity. Even when the independence of matrix entries is dropped and local correlations with sufficient decay are considered this classification persists [6] and concentration of the spectral measure has been proven in broad generality [10, 13, 19, 26, 30, 35, 41, 43, 46] Hermitian matrices since their spectral instability makes such questions more challenging compared to the Hermitian situation. To resolve the instability we perform a non-linear transformation of the MDE that allows to restrict the analysis to the manifold of perturbations that respect the additional symmetry of M This transformation is applicable in the context of other non-normal models, e.g. non-Hermitian polynomials in several non-commutative variables. It is crucial to show that σ is a real analytic function of |ζ|2
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