The circular law for random band matrices: improved bandwidth for general models

TL;DR

This paper establishes the convergence of the empirical spectral distribution of non-Hermitian random band matrices to the circular law for broader bandwidth regimes, improving previous thresholds and handling general variance profiles.

Contribution

It proves the circular law for random band matrices with doubly stochastic variance profiles at lower bandwidth thresholds than previously known, for both Gaussian and subgaussian entries.

Findings

Circular law holds for $eta > 5/6$ with Gaussian entries.

Circular law holds for $eta > 8/9$ with subgaussian entries.

Extended product circular law with growing matrix number.

Abstract

We consider the convergence of the ESD for non-Hermitian random band matrices with independent entries to the circular law, which is the uniform measure on the unit disk in the center of the complex plane. We assume that the bandwidth of the matrix scales like $n^\gamma$ for some $\gamma\in(0,1]$, where $n$ is the matrix size, and the variance profile of the matrix is only assumed to be doubly stochastic with no additional assumption on its specific mixing properties. We prove that the circular law limit holds either (1) when $\gamma>\frac{5}{6}$ and the entries are independent Gaussians, (2) or when $\gamma>\frac{8}{9}$ and the entries are independent subgaussian random variables. This new threshold improves the previous threshold $\gamma>\frac{32}{33}$ which was only proven for block band matrices and periodic band matrices. After the initial version of this paper, the author further extended the range of circular law for much smaller values of $\gamma$ in 2508.18143 and 2511.01744 when the variance profile has specific mixing properties, but not for an arbitrary doubly stochastic variance profile. Thus the main contribution of this paper is the circular law for a genuine power law bandwidth for any doubly stochastic variance profile. We also prove an extended form of product circular law with a growing number of matrices. Weak delocalization estimates on eigenvectors are also derived. The new technical input is new polynomial lower bounds on some intermediate small singular values, and this estimate does not depend on the specific structure of the variance profile beyond the fact that it is doubly stochastic.