On pseudospectrum of inhomogeneous non-Hermitian random matrices

2023/07/17 by Tikhomirov, Konstantin · 4 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2307.08211

Abstract

Let A be an n× n matrix with mutually independent centered Gaussian entries. Define σ^*:=max_{i,j≤ n}√\mathbb E |A_i,j|², σ:=max(max_{j≤ n}√\mathbb E ‖\rm col_j(A)‖₂², max_{i≤ n}√\mathbb E ‖\rm row_i(A)‖₂²). Assume that σ≥ n^ε σ^* for a constant ε>0, and that a complex number z satisfies |z|=Ω(σ). We prove that s_min(A-z \rm Id) ≥ |z| exp(-n^o(1) (\frac√(n) σ^*σ)²) with probability 1-o(1). Without extra assumptions on A, the bound is optimal up to the n^o(1) multiple in the power of exponent. We discuss applications of this estimate in context of empirical spectral distributions of inhomogeneous non-Hermitian random matrices.

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