Hopf algebras and Markov chains: Two examples and a theory

TL;DR

This paper explores how operations in combinatorial Hopf algebras induce Markov chains, providing explicit descriptions of eigenvectors, quasi-stationary distributions, and absorption rates for various processes like card shuffling and rock-breaking.

Contribution

It introduces a unified algebraic framework linking Hopf algebra operations to Markov chains, enabling explicit analysis of their spectral properties and convergence behaviors.

Findings

Explicit eigenvector descriptions for card shuffling.

Quasi-stationary distribution and absorption rates for rock-breaking.

Unified algebraic approach to analyze diverse Markov chains.

Abstract

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.