Random Walks on $\mathbb{Z}_q^d$

TL;DR

This paper analyzes long-range random walks on finite and infinite lattice structures, exploring their spectral properties, eigenfunctions, and limiting behaviors, with applications to mixing times and central limit theorems.

Contribution

It introduces spectral expansions for random walks on $ ext{Z}_q^d$, utilizing multivariate Krawtchouk polynomials, and extends the analysis to the torus with limit theorems as dimension grows.

Findings

Spectral decomposition using multivariate Krawtchouk polynomials.

Limit theorems for transition distributions as dimension increases.

Analysis of cutoff and mixing times for these processes.

Abstract

This paper studies long range random walks on ${\mathbb{Z}_q}^d$. $X_{t+1} = X_t + Z_t \mod q$, with $(Z_t)$ independent and identically distributed. Multiple entries of $Z_t$ can be non-zero in a transition. An emphasis is on finding the structure of such random walks and spectral expansions for the transition functions. Circulant transition probability matrices are important in this study. Processes are extended to processes on the torus $[0,1]$, scaling entries in $\{0,1,\ldots, q-1\}$ by dividing by $q$ and letting $q \to \infty$. If the entries of $X_t$ are exchangeable then a grouping of $X_t$ is made by taking counts of the types $0,\ldots, q-1$ in $X_t$. In this grouping the multivariate Krawtchouk polynomials become the eigenvectors. Examples consider cutoff times and mixing times in these processes. A limit form for the multivariate Krawtchouk polynomials is used to find a central limit theorem for the transition distributions in the grouped model as $d \to \infty$.