Total variation cutoff in a tree

TL;DR

This paper constructs specific trees where lazy simple random walks exhibit total variation cutoff, analyzing the hitting times, mixing time, relaxation time, and cutoff window to understand the phenomenon.

Contribution

It introduces a new family of trees demonstrating total variation cutoff and provides detailed analysis of their mixing properties.

Findings

Identified trees with total variation cutoff behavior.

Computed mixing time, relaxation time, and cutoff window for these trees.

Showed hitting times of large sets are concentrated around their means.

Abstract

We construct a family of trees on which a lazy simple random walk exhibits total variation cutoff. The main idea behind the construction is that hitting times of large sets should be concentrated around their means. For this sequence of trees we compute the mixing time, the relaxation time and the cutoff window.