Invertibility for non-Hermitian and symmetric random band matrices with sublinear bandwidth and discrete entries

TL;DR

This paper investigates the invertibility of non-Hermitian and symmetric random band matrices with sublinear bandwidth, establishing high-probability invertibility results for certain distributions and extending known results to more general models.

Contribution

It provides the first high-probability invertibility results for a broad class of sublinear bandwidth random band matrices, including symmetric and non-Hermitian cases.

Findings

Invertibility with probability $1- ext{exp}(- ext{Omega}(n^{rac{ ext{alpha}}{2}}))$ for $ ext{alpha}>rac{2}{3}$.

Results cover models with variance profiles like block and periodic band matrices.

Extension of invertibility results to symmetric matrices with integer entries.

Abstract

A well-known result in random matrix theory, proven by Kahn, Koml\'os and Szemer\'edi in 1995, states that a square random matrix with i.i.d. uniform $\{\pm 1\}$ entries is invertible with probability $1-\exp(-\Omega(n))$. As a natural generalization of the model, we consider the invertibility of a class of random band matrices with independent entries where the bandwidth $d_n$ scales like $n^\alpha$, for some $\alpha\in(0,1)$. The band matrix model we consider is sufficiently general and covers existing models such as the block band matrix and periodic band matrix, allowing great flexibility in the variance profile. As the bandwidth is sublinear in the dimension, estimating the invertibility and least singular values of these matrices is a well-known open problem. We make progress towards the invertibility problem by showing that, when $\alpha>\frac{2}{3}$ and when the random variables are i.i.d. uniformly distributed on $\{\pm 1+c\}$ for any fixed integer $c$, then the band matrix is invertible with probability $1-\exp(-\Omega(n^{\alpha/2}))$. Previously, even invertibility with probability $1-o(1)$ was not known for these band matrix models except in the very special case of block band matrices. We then extend the invertibility result to symmetric random band matrices with integer entries, and prove the same non-singularity probability estimate whenever $\alpha>\frac{2}{3}$.