Exponential phi-mixing implies exponential psi-mixing for Markov fields on bounded degree graphs

TL;DR

This paper demonstrates that for certain Markov fields on bounded degree graphs, exponential phi-mixing guarantees exponential psi-mixing, with applications to Gibbs fields like the Ising and Potts models at low temperatures.

Contribution

It establishes a new link between phi-mixing and psi-mixing for Markov fields on graphs, with explicit decay rate relations and applications to statistical physics models.

Findings

Exponential phi-mixing implies exponential psi-mixing under specified conditions.

Derived explicit decay rate bounds for psi-mixing from phi-mixing rates.

Applied results to Gibbs fields on regular trees, including Ising and Potts models.

Abstract

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 $(\lambda - 3\theta)/9$. 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.