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The cutoff phenomenon describes a case where a Markov chain exhibits a sharptransition in its convergence to stationarity. In 1996, Diaconis surveyed thisphenomenon, and asked how one could recognize its occurrence in families offinite ergodic Markov chains. In 2004, the third author noted that a necessarycondition for cutoff in a family of reversible chains is that the product ofthe mixing-time and spectral-gap tends to infinity, and conjectured that inmany settings, this condition should also be sufficient. Diaconis andSaloff-Coste (2006) verified this conjecture for continuous-timebirth-and-death chains, started at an endpoint, with convergence measured inseparation. It is natural to ask whether the conjecture holds for these chainsin the more widely used total-variation distance.  In this work, we confirm the above conjecture for all continuous-time or lazydiscrete-time birth-and-death chains, with convergence measured viatotal-variation distance. Namely, if the product of the mixing-time andspectral-gap tends to infinity, the chains exhibit cutoff at the maximalhitting time of the stationary distribution median, with a window of at mostthe geometric mean between the relaxation-time and mixing-time.  In addition, we show that for any lazy (or continuous-time) birth-and-deathchain 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 inany family of such chains.

Total-variation cutoff in birth-and-death chains