Local limit theorems for random walks on a large discrete torus

TL;DR

This paper establishes local limit theorems for random walks on large discrete tori, considering different regimes based on bandwidth, walk length, and torus size, inspired by random matrix edge statistics.

Contribution

It introduces new local limit theorems for random walks on large tori with transition matrices from discretized densities, covering multiple regimes.

Findings

Proved local limit theorems in three regimes

Analyzed the impact of bandwidth, walk length, and torus size

Connected random walk behavior to random matrix edge statistics

Abstract

Inspired by the study of edge statistics of random band matrices, we investigate random walks on large $d$-dimensional periodic lattices, whose transition matrices are determined by discretized density functions. Under certain moment assumption on the density, we prove local limit theorems for random walks in three different regimes according to the bandwidth parameter, random walk length and torus size.