The pressure, densities and first order phase transitions associated with multidimensional SOFT

TL;DR

This paper investigates the mathematical properties of the pressure function in multidimensional Potts models, establishing its convexity and Lipschitz continuity, and linking phase transitions to points of non-differentiability, with computational methods applied to specific models.

Contribution

The paper provides rigorous proofs of pressure properties and develops computable bounds, applying them to confirm and extend previous heuristic results for the monomer-dimer model.

Findings

Pressure is Lipschitz and convex.

First order phase transitions occur at non-differentiable points of pressure.

Numerical methods accurately compute pressure and density entropy.

Abstract

We study theoretical and computational properties of the pressure function for subshifts of finite type on the integer lattice $\Z^d$, multidimensional SOFT, which are called Potts models in mathematical physics. We show that the pressure is Lipschitz and convex. We use the properties of convex functions to show rigorously that the phase transition of the first order correspond exactly to the points where the pressure is not differentiable. We give computable upper and lower bounds for the pressure, which can be arbitrary close the values of the pressure given a sufficient computational power. We apply our numerical methods to confirm Baxter's heuristic computations for two dimensional monomer-dimer model, and to compute the pressure and the density entropy as functions of two variables for the two dimensional monomer-dimer model.