Exponential phi-mixing implies exponential psi-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 $\theta$ uniform $\phi$-mixing with exponential decay rate $\lambda >3\theta$ implies uniform $\psi$-mixing with exponential decay rate $(\lambda -3\theta )\mathrm{/}9$. As an application we obtain exponential $\psi$-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.