Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

Abstract

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists τ ∈ (0, 1) so that if the bandwidth of the matrix X is at least n 1−τ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of X, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting [26].

Quantitative invertibility of non-Hermitian random matrices

“…where C > 0 is a universal constant. Invertibility of structured random matrices and applications to the study of limiting spectral distribution have been considered, in particular, in [75,17,19,20,39]. A basic model of interest here is of the form A n = U n ⊙ M n − z Id, where M n is a matrix with i.i.d entries of zero mean and unit variance, z ∈ C is some complex number, U n is a non-random matrix with non-negative real entries encoding the standard deviation profile, and "⊙" denotes the Hadamard (entry-wise) product of matrices.…”

“…One of the results of [19] is the circular law for doubly stochastic variance profiles: The setting of sparse structured matrices is not well understood. For results in that direction, we refer to a recent paper [39] dealing with invertibility and spectrum of block band matrices.…”