A sharp analysis of the mixing time for random walk on rooted trees

TL;DR

This paper introduces a new measure for rooted trees, analyzes a Markov chain's mixing time, and proves that approximately n^2 steps are needed for convergence, using combinatorial methods.

Contribution

It defines an analog of Plancherel measure for rooted trees and establishes the mixing time of the associated Markov chain.

Findings

Order n^2 steps are necessary and sufficient for convergence.

The measure is an analog of Plancherel measure for rooted trees.

Combinatorics of commutation relations is used in the analysis.

Abstract

We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.