Random walks on rings and modules

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 ∈ R, b ∈ 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).