The topological strong spatial mixing property and new conditions for pressure approximation

TL;DR

This paper introduces the topological strong spatial mixing (TSSM) property as a new combinatorial condition that ensures efficient pressure approximation in Gibbs measures with strong spatial mixing, extending previous theoretical results.

Contribution

The paper defines TSSM, proves its sufficiency and necessity for strong spatial mixing, and extends existing pressure representation results to systems with hard constraints.

Findings

TSSM is sufficient for efficient pressure approximation.

TSSM is necessary for high-rate strong spatial mixing.

Extended pressure representation results to constrained systems.

Abstract

In the context of stationary $\mathbb{Z}^d$ nearest-neighbour Gibbs measures $\mu$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property (TSSM)) on the support of $\mu$ sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of TSSM for studying strong spatial mixing on systems with hard constraints. We also show that TSSM is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by D. Gamarnik and D. Katz (2009), and B. Marcus and R. Pavlov (2013), who gave a special representation of topological pressure in terms of conditional probabilities.