Note: Random-to-front shuffles on trees

TL;DR

This paper analyzes a Markov chain on tree orderings with local random-to-front transitions, deriving eigenvalues using semigroup theory, providing insights into the chain's spectral properties.

Contribution

It introduces a novel Markov chain model on tree orderings and applies Brown's semigroup theory to determine its eigenvalues, advancing understanding of such stochastic processes.

Findings

Eigenvalues of the transition matrix are explicitly determined.

The Markov chain's spectral properties are characterized.

The model extends random-to-front shuffles to tree structures.

Abstract

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.