Random walks on rings and modules

TL;DR

This paper analyzes two models of random walks on modules over finite commutative rings, deriving their transition spectra using representation theory, which advances understanding of algebraic random processes.

Contribution

It provides the complete spectral characterization of these random walks on modules over finite rings using representation theory.

Findings

Spectral analysis of coin-toss and affine walks on modules.

Complete transition matrix spectra derived from monoid representations.

Applicable to arbitrary measures on the ring R.

Abstract

We consider two natural models of random walks on a module $V$ over a finite commutative ring $R$ driven simultaneously by addition of random elements in $V$, and multiplication by random elements in $R$. In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements $a \in R,b \in V$ are sampled independently, and the current state $x$ is taken to $ax+b$. For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on $V$ under a suitable hypothesis on the measure on $V$ (the measure on $R$ can be arbitrary).