Computing cutoff times of birth and death chains

TL;DR

This paper develops methods to explicitly compute cutoff times and windows for birth and death chains, complementing existing spectral and total variation criteria, with applications to various examples.

Contribution

It introduces new techniques for calculating cutoff times and windows, extending previous spectral and total variation criteria for birth and death chains.

Findings

Computed cutoff times and windows for multiple examples

Extended spectral and total variation criteria

Provided practical methods for analyzing convergence

Abstract

Earlier work by Diaconis and Saloff-Coste gives a spectral criterion for a maximum separation cutoff to occur for birth and death chains. Ding, Lubetzky and Peres gave a related criterion for a maximum total variation cutoff to occur in the same setting. Here, we provide complementary results which allow us to compute the cutoff times and windows in a variety of examples.