Poincar\'e inequality for Markov random fields via disagreement percolation

We consider Markov random fields of discrete spins on the lattice $\text{Zd}$. We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is “sufficiently” subcritical, then we prove the Poincar'e inequality. In the whole subcritical regime, we have a weak Poincar'e inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.