A Class of Long Range Circulant Random Walks on $\mathbb{Z}_q^d$

Abstract

This paper studies a class of long range random walks $(X_t)_{t=0}^\infty$ on the direct product of cyclic groups $\mathbb{Z}_q^d$ for $d\ge 1$ and $q\ge 2$. $X_{t+1} = X_t + Z_t \mod q$, with $(Z_t)_{t=1}^\infty$ \iid on $\{0,1,\ldots, q-1\}^d$. Entries of $Z_t$ are updated by circulant matrices, possibly with dependence. 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. An extension is made to processes on a $d$-dimensional torus, scaling entries in $\{0,1,\ldots, q-1\}$ by dividing by $q$ and letting $q\to \infty$. The state space is then $d$ circles of unit perimeter, where $0$ and $1$ are identified as the same point in each circle. 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 $d \to \infty$.