Algorithmic Pirogov-Sinai theory

TL;DR

This paper introduces an efficient algorithmic framework combining Pirogov-Sinai contour methods with Barvinok's series approach to approximate counting and sampling in low-temperature statistical physics models on lattices and tori.

Contribution

It develops a novel algorithmic approach that integrates contour representations with Taylor series truncation for efficient approximation in statistical physics models.

Findings

Provides an FPTAS for the hard-core model's partition function at high fugacity.

Develops an efficient sampling algorithm for the ferromagnetic Potts model at low temperature.

Achieves approximation and sampling in regimes previously computationally challenging.

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.