Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles (Q6616881)

This work establishes general quantitative convergence bounds (spectral gap and mixing time) for discrete-time continuous-space reversible Markov chains, based on the isoperimetry profile of their invariant measure, which relates the measure of sets and of their boundary, and a so-called closed coupling condition, which states that there is a distance at which two chains can be coupled with a given probability. This framework is particularly well adapted to the Metropolis-Hastings chain. It is applied to the Random Walk Monte Carlo chain for log-concave targets, providing sharp explicit upper and lower bounds on the spectral gap. The approach is actually able to take into account that, for initial times, the convergence is in fact faster than the spectral gap alone would give, which leads to good estimates for mixing times from feasible starts. The same analysis is conducted for the preconditioned Crank-Nicolson algorithm.