Non-Hermitian random matrices with a variance profile (II): properties\n and examples

2020/07/28 by Nicholas A. Cook, Walid Hachem, Cook, Nicholas A. +5 · 3 citations
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2007.15438

openalex publication_date 2020/07/28 · openalex created_date 2021/12/06 · openalex updated_date 2026/07/28

Abstract

For each n, let A_n=(\σ_ij) be an n\× n deterministic matrix\nand let X_n=(X_ij) be an n\× n random matrix with i.i.d. centered\nentries of unit variance. In the companion article Cook et al., we considered\nthe empirical spectral distribution \μ_n^Y of the rescaled entry-wise\nproduct \Y_n =
frac 1
sqrtn A_n
odot X_n =
left(
frac1
sqrtn\n
sigma_ijX_ij
right) and provided a deterministic sequence of\nprobability measures \μ_n such that the difference \μ^Y_n - \μ_n\nconverges weakly in probability to the zero measure. A key feature in Cook et\nal. was to allow some of the entries \σ_ij to vanish, provided that the\nstandard deviation profiles A_n satisfy a certain quantitative irreducibility\nproperty.\n In the present article, we provide more information on the sequence\n(\μ_n), described by a family of Master Equations. We consider these\nequations in important special cases such as separable variance profiles\n\σ²_ij=d_i widetilde d_j and sampled variance profiles \σ²_ij\n= \σ²\( frac in, frac jn \) where (x,y)\↦ \σ²(x,y)\nis a given function on [0,1]². Associate examples are provided where\n\μ_n^Y converges to a genuine limit.\n We study \μ_n's behavior at zero and provide examples where \μ_n's\ndensity is bounded, blows up, or vanishes while an atom appears. As a\nconsequence, we identify the profiles that yield the circular law.\n Finally, building upon recent results from Alt et al., we prove that except\nmaybe in zero, \μ_n admits a positive density on the centered disc of radius\n\√(\ρ(V_n)), where V_n=( frac 1n \σ_ij²) and \ρ(V_n) is its\nspectral radius.\n

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• The circular law for non-Hermitian random band matrices up to bandwidth N^1/2+c
• Convergence and asymptotic freeness of missing data matrices
• Invertibility for non-Hermitian and symmetric random band matrices with sublinear bandwidth and discrete entries

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