Exponential phi-mixing implies exponential psi-mixing for Markov fields on bounded degree graphs
Elias Zimmermann
TL;DR
The paperAddress the problem of relating exponential φ-mixing to exponential ψ-mixing for non-degenerate $k$-Markovian random fields on graphs with bounded degree and exponential growth. It proves a main theorem: whenever the field is exponential φ-mixing with rate $\lambda$ and $\lambda > 3\theta$, where $\theta$ is the exponential growth rate of the graph, the field is exponential ψ-mixing with rate $(\lambda - 3\theta)/9$, using a coupling-based, growth-aware analysis that extends Alexander’s lattice approach to general graphs. The results yield explicit decay-rate bounds and apply to Gibbs fields on regular trees, including Ising models at low temperature and large-state Potts models, under suitable Dobrushin-type conditions. This broadens the scope of exponential mixing results beyond square lattices and provides practical rate estimates for statistical mechanics models on tree-like and other bounded-degree graphs, enabling LLN/CLT-type conclusions for dependent fields.
Abstract
We show that for non-degenerate $k$-Markovian random fields with finite state space over a bounded degree graph with exponential growth rate $θ$ uniform $φ$-mixing with exponential decay rate $λ> 3θ$ implies uniform $ψ$-mixing with exponential decay rate $(λ- 3θ)/9$. As an application we obtain exponential $ψ$-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.