Markov processes on a circular lattice

Sourav Majumdar

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arXiv · 2603.02890

Markov processes on a circular lattice

Sourav Majumdar

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Abstract

We develop a Markov process viewpoint for discrete circular distributions motivated by directional-statistics settings where angles are observed on a finite grid and evolve over time. On the $m$-point discrete circle, the cycle graph, we study diffusion-generated families, obtaining an explicit transition kernel, exact trigonometric moments, and convergence to uniformity. We present a simple approach to construct reversible nearest-neighbour chains with any prescribed strictly positive stationary pmf $\pi$, providing discrete analogues of Markov processes on the continuous circle. We construct processes whose stationary laws are the discrete von Mises and wrapped Cauchy distributions with closed-form normalizers and exact moments.

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Sourav Majumdar. 2026-03-03. Markov processes on a circular lattice. https://arxiv.org/abs/2603.02890

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