Exponential \(\Phi \)-Mixing Implies Exponential \(\Psi \)-Mixing For Markov Fields On Bounded Degree Graphs

Abstract

We show that for non-degenerate k-Markovian random fields with finite state space over a bounded degree graph with exponential growth rate , uniform -mixing with exponential decay rate implies uniform -mixing with exponential decay rate . As an application we obtain exponential -mixing for Gibbs fields on regular trees arising from finite range potentials such as the Ising model at low inverse temperature or the Potts model with sufficiently many spin states.