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Coupling, concentration inequalities and stochastic dynamics

In the context of interacting particle systems, we study the influence of the action of the semigroup on the concentration property of Lipschitz functions. As an application, this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems. We illustrate our approach in a variety of examples for which we obtain several new results with short and non-technical proofs. These examples include the symmetric and asymmetric exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics"). We also give a new proof of the Poincaré inequality, based on coupling, in the context of one-dimensional Gibbs measures. In particular, we cover the case of polynomially decaying potentials, where the log-Sobolev inequality does not hold.