Exponential Φ-Mixing Implies Exponential Ψ-Mixing For Markov Fields On Bounded Degree Graphs

We show that for non-degenerate k -Markovian random fields with finite state space over a bounded degree graph with exponential growth rate "Equation missing" , uniform "Equation missing" -mixing with exponential decay rate "Equation missing" implies uniform "Equation missing" -mixing with exponential decay rate "Equation missing" . As an application we obtain exponential "Equation missing" -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.