Total-variation cutoff in birth-and-death chains

TL;DR

This paper proves that for birth-and-death Markov chains, the total-variation cutoff phenomenon occurs if and only if the product of the mixing-time and spectral-gap tends to infinity, confirming a conjecture for these chains.

Contribution

It establishes the equivalence of total-variation cutoff and separation cutoff for birth-and-death chains, extending previous results to a broader setting.

Findings

Cutoff occurs at the maximal hitting time of the median stationarity.

The cutoff window is at most the geometric mean of relaxation and mixing times.

Total-variation cutoff is equivalent to separation cutoff in these chains.

Abstract

The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time birth-and-death chains, started at an endpoint, with convergence measured in separation. It is natural to ask whether the conjecture holds for these chains in the more widely used total-variation distance. In this work, we confirm the above conjecture for all continuous-time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time. In addition, we show that for any lazy (or continuous-time) birth-and-death chain with stationary distribution $\pi$, the separation $1 - p^t(x,y)/\pi(y)$ is maximized when $x,y$ are the endpoints. Together with the above results, this implies that total-variation cutoff is equivalent to separation cutoff in any family of such chains.