Unified theory for finite Markov chains

TL;DR

This paper introduces a unified, algebraic framework using geometric semigroup theory to compute stationary distributions of finite irreducible Markov chains without linear algebra, generalizing previous models like the Tsetlin library.

Contribution

It develops a novel approach applying semigroup expansions and normal forms to analyze Markov chains, extending prior work on special semigroup classes.

Findings

Provides formulas for stationary distributions using normal forms and Kleene expressions.

Estimates mixing times based on normal form analysis.

Generalizes previous models like the Tsetlin library to broader classes of Markov chains.

Abstract

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.