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

TL;DR

This paper establishes Poincaré inequalities for Markov random fields on lattices using disagreement percolation, providing bounds on relaxation times in subcritical regimes.

Contribution

It introduces a coupling technique for conditional distributions to prove Poincaré inequalities in subcritical regimes of Markov random fields.

Findings

Poincaré inequality holds when disagreement clusters are subcritical

Weak Poincaré inequality in the entire subcritical regime

Polynomial bounds for Glauber dynamics relaxation times

Abstract

We consider Markov random fields of discrete spins on the lattice $\Zd$. 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.