The product condition and cutoff for non-reversible semi-birth-and-death chains

The cutoff phenomenon describes an abrupt transition in the convergence of a finite Markov chain to its stationary distribution. For reversible chains, cutoff requires the relaxation time to be negligible compared with the mixing time, a requirement known as the product condition. Basu, Hermon and Peres proved that this condition is also sufficient for reversible semi-birth-and-death chains. This thesis establishes a continuous-time, non-reversible counterpart of their result. We consider (δ, r)-semi-birth-and-death chains, whose jumps have size at most r and whose nearest-neighbour transition probabilities are bounded below by δ. In place of the usual reversible relaxation time, we use the singular-value-based relaxation time introduced by Chatterjee. The proof relates mixing to the hitting time of a central block. We show that the chain mixes rapidly after reaching this block and that the time required to reach it is sharply concentrated. It follows that whenever Chatterjee’s relaxation time is negligible compared with the mixing time, the sequence exhibits total-variation cutoff.