Random motion on finite rings, II: Noncommutative rings

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.