Semigroups, rings, and Markov chains

TL;DR

This paper studies random walks on left-regular band semigroups, including hyperplane walks and matroid-related walks, providing eigenvalue formulas and applications to combinatorial structures.

Contribution

It introduces a ring-theoretic approach to analyze eigenvalues of semigroup-based Markov chains, including explicit formulas and new examples.

Findings

Transition matrices are diagonalizable with explicitly computed eigenvalues.

Constructs new random walks on distributive lattices and matroids.

Eigenvalue multiplicities relate to generalized derangement numbers.

Abstract

We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are ``generalized derangement numbers'', which may be of independent interest.