Random motion on finite rings, I: commutative rings

TL;DR

This paper studies Markov chains on finite commutative rings generated by random addition and multiplication, deriving eigenvalues, stationary distributions, and mixing times, with explicit results for specific ring classes.

Contribution

It provides new formulas for eigenvalues, stationary probabilities, and mixing times of Markov chains on finite rings, including explicit solutions for chain rings.

Findings

Eigenvalues and multiplicities formulas derived using character theory.

Explicit stationary distribution formulas for finite chain rings.

Proven constant mixing time using coupling methods.

Abstract

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first prove formulas for eigenvalues and multiplicities of the transition matrices of these chains using the character theory of finite abelian groups. The examples of principal ideal rings (such as $\mathbb{Z}_{n}$) and finite chain rings (such as $\mathbb{Z}_{p^k}$) are particularly illuminating and are treated separately. We then prove a recursive formula for the stationary probabilities for any ring, and use it to prove explicit formulas for the probabilities for finite chain rings when multiplication is also uniformly random. Finally, we prove constant mixing time for our chains using coupling.