Poincaré inequalities for Markov chains: a meeting with Cheeger, Lyapunov and Metropolis

We develop a theory of weak Poincar\é inequalities to characterize convergence rates of ergodic Markov chains. Motivated by the application of Markov chains in the context of algorithms, we develop a relevant set of tools which enable the practical study of convergence rates in the setting of Markov chain Monte Carlo methods, but also well beyond.