Random motion on finite rings, II: Noncommutative rings

TL;DR

This paper generalizes the analysis of Markov chains on finite rings to include noncommutative rings, providing explicit eigenvalues, stationary distributions, and mixing times, with detailed results for matrix rings over finite fields.

Contribution

It extends previous work from commutative to noncommutative finite rings, deriving explicit spectral formulas and stationary distributions for these more complex algebraic structures.

Findings

Eigenvalues of transition matrices are explicitly characterized.

Stationary distributions are computed recursively.

Mixing times are bounded by a universal constant.

Abstract

We extend our previous study of Markov chains on finite commutative rings (arXiv:1605.05089) to arbitrary finite rings with identity. At each step, we either add or multiply by a randomly chosen element of the ring, where the addition (resp. multiplication) distribution is uniform (resp. conjugacy invariant). We prove explicit formulas for some of the eigenvalues of the transition matrix and give lower bounds on their multiplicities. We also give recursive formulas for the stationary distribution and prove that the mixing time is bounded by an absolute constant. For the matrix rings $M_2(\mathbb F_q),$ we compute the entire spectrum explicitly using the representation theory of $\text{GL}_2(\mathbb F_q),$ as well as the stationary probabilities.