# 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.

## Key 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.