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

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.