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

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 $d \to \infty$.