Cutoff on trees is rare

TL;DR

This paper investigates the cutoff phenomenon in simple random walks on trees, providing geometric criteria to determine when cutoff occurs or is absent, with applications to various tree families.

Contribution

It introduces easy-to-verify geometric criteria for cutoff in random walks on trees, extending understanding to diverse tree structures.

Findings

Cutoff occurs in certain tree families based on geometric conditions.

No cutoff in spherically symmetric and Galton-Watson trees under specified conditions.

Criteria help identify cutoff presence or absence in complex tree models.

Abstract

We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine whether the cutoff phenomenon occurs. We thoroughly discuss families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton-Watson trees of a fixed height, and sequences of random trees converging to the Brownian CRT.