The Poincaré inequality for Markov random fields proved via disagreement percolation

We consider Markov random fields of discrete spins on the lattice ZdZd. We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is “sufficiently” subcritical, then we are able to prove the Poincaré inequality. For the whole subcritical regime, we have a weak Poincaré inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.