No cutoff in Spherically symmetric trees

TL;DR

This paper proves that lazy simple random walks on finite spherically symmetric trees have a bounded ratio of mixing to relaxation time, do not exhibit pre-cutoff, and have non-concentrated hitting times, with stability under rough isometries.

Contribution

It establishes universal bounds on mixing and relaxation times, answers a recent open question, and analyzes hitting time concentration and stability under rough isometries for these trees.

Findings

Bounded ratio of mixing to relaxation time

No pre-cutoff for lazy simple random walks

Hitting times are non-concentrated

Abstract

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random walks. This answers a question recently proposed by Gantert, Nestoridi, and Schmid. We also show that for lazy simple random walks on finite spherically symmetric trees, hitting times of vertices are (uniformly) non concentrated. Finally, we study the stability of our results under rough isometries.