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).

The quiver of an affine monoid

“…Ayyer and the second author described the simple C Aff(R)-modules [4]. Since we only need the case of a local ring R, we specialize to this case.…”

“…In a recent paper [4], Ayyer and the second author studied Markov chains coming from random applications of ax + b mappings on a finite commutative ring R using the representation theory of the affine monoid Aff(R) of all such maps. Using this theory, they were able to compute the spectrum of the transition matrix of the random walk under certain restrictions on the underlying probability measure.…”

“…In this paper we compute the quiver of C Aff(R) for any finite commutative ring R. It was observed in [4] that the idempotents of Aff(R) form a left regular band, that is, a monoid of idempotents in which each principal right ideal has a unique generator. Such monoids belongs to a class of monoids whose quivers are described in a paper of Margolis and the second author [15].…”

Random Motion on Finite Rings, I: Commutative Rings

“…(b) For any A ∈ 0 1 0 0 1 z 0 0 z∈Fq and ρ ∈ Σ A , we have; 4) and each eigenvalue above appears with multiplicity (q + 1) dim(ρ).…”

“…In an earlier work, we have studied a general class of random walks on finite commutative rings [3]. This has since been extended to random walks on modules of finite commutative rings [4].…”