Cutoff in total variation for birth-and-death chains
Seminar Room 1, Newton Institute
We prove that a family of continuous-time or lazy discrete-time birth-and-death chains, exhibits the cutoff phenomenon if and only if the product of the mixing-time and spectral-gap tends to infinity; in this case, the cutoff window of at most the geometric mean between the relaxation-time and mixing-time. An analogous result for convergence in separation was proved earlier by Diaconis and Saloff-Coste (2006) for birth-and-death chains started at an endpoint; 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. (Joint with J. Ding and Y. Peres).
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