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Local limit theorems for random walks on a large discrete torus

Yandong Gu, Dang-Zheng Liu

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.

Local limit theorems for random walks on a large discrete torus

Abstract

Inspired by the study of edge statistics of random band matrices, we investigate random walks on large -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.

Paper Structure

This paper contains 8 sections, 6 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Proofs of Theorem \ref{['pllt']} and Proposition \ref{['p-uub']}
  5. One-step transition probability
  6. Proof of Theorem \ref{['pllt']}
  7. Proof of Proposition \ref{['p-uub']}
  8. Proof of Theorem \ref{['LLT']}

Key Result

Theorem 1.3

For the random walk given in Definition def-CRM, assume that $f$ is continuous at $x_0$ and $f(x_0)>0$ for some fixed $x_0\in [0,\frac{L}{2W})^d$, and also where $\alpha \in (0,2)$. As $L\to \infty$ and $W\to\infty$, the following three types of local limits hold.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['pllt']}
  • proof : Proof of Proposition \ref{['p-uub']}
  • ...and 5 more