Algorithmic Pirogov-Sinai theory

Abstract

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \mathbb Z)^d$. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $\mathbb Z^d$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $(\mathbb Z/n \mathbb Z)^d$ at sufficiently low temperature.