Random Walks on $\mathbb{Z}_q^d$
Robert Griffiths, Shuhei Mano
TL;DR
The paper develops a unified spectral framework for long-range random walks on ${\mathbb Z}_q^d$, where increments $Z_t$ may affect multiple coordinates. It first analyzes circulant transition matrices on ${\cal V}_q$ and derives explicit eigenstructures, linking eigenvalues to random increments via $\eta_r=\mathbb{E}[\theta_r^V]$ and establishing Lancaster-type characterizations. Extending to ${\cal V}_{q,d}$, it shows that transition kernels admit eigenvectors of the form $\theta_1^{x\cdot r}$ with eigenvalues $\rho_r=\mathbb{E}[\theta_1^{V\cdot r}]$, and, under exchangeability, yields a spectral expansion in multivariate Krawtchouk polynomials for the grouped counts $M_t$, together with a central limit form as $d\to\infty$. The work also develops a torus-limit process on $[0,1]$ with Fourier eigenfunctions, and treats de Finetti-type models that produce mixtures of increments, providing cutoff and mixing-time insights. Overall, the article unifies algebraic and probabilistic methods to analyze spectral expansions, mixing behavior, and limit theorems for high-dimensional, long-range Markov processes on discrete and continuous state spaces.
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 $d \to \infty$.