Unified theory for finite 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 KarnofskyRhodes 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 setup generalizes previous groundbreaking work involving leftregular bands (or $\mathscr{R}$trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The KarnofskyRhodes 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 arXiv:
 arXiv:1711.10689
 Bibcode:
 2017arXiv171110689R
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 20M30;
 60J10 (Primary) 20M05;
 60B15;
 60C05 (Secondary)
 EPrint:
 29 pages, 12 figures