Time-inhomogeneous random walks on finite groups and cokernels of random integer block matrices

TL;DR

This paper investigates the convergence behavior of time-inhomogeneous random walks on finite groups with non-generating supports and applies the results to universality in cokernels of dependent random integer matrices.

Contribution

It introduces conditions under which such random walks converge to uniform distribution and applies spectral analysis to establish universality results for cokernels of dependent random matrices.

Findings

Random walks converge to uniform distribution under certain support conditions.

Spectral bounds provide convergence rate estimates.

Universality of cokernels extends to matrices with dependent entries.

Abstract

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we use the moment method of Wood to prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.