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 $\hat{I}¸$ uniform $\ddot{I}•$-mixing with exponential decay rate $\hat{I}»>3\hat{I}¸$ implies uniform $\ddot{I}ˆ$-mixing with exponential decay rate $(\hat{I}»-3\hat{I}¸)/9$. As an application we obtain exponential $\ddot{I}ˆ$-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.