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

In the context of stationary nearest-neighbour Gibbs measures satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property) on the support of that is sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of topological strong spatial mixing for studying strong spatial mixing on systems with hard constraints. We also show that topological strong spatial mixing is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by Gamarnik and Katz [Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys. 137(2) (2009), 205-232], and Marcus and Pavlov [An integral representation for topological pressure in terms of conditional probabilities. Israel J. Math. 207(1) (2015), 395-433], who gave a special representation of topological pressure in terms of conditional probabilities.