Pirogov--Sinai Theory Beyond Lattices

TL;DR

This paper extends Pirogov--Sinai theory, traditionally used for lattice models, to more general combinatorial settings, specifically applying it to the hard-core model of independent sets beyond lattice structures.

Contribution

It develops a flexible combinatorial framework for Pirogov--Sinai theory applicable to non-lattice models, introducing new contour definitions based on cycle basis connectivity.

Findings

Main conclusions of Pirogov--Sinai theory hold in broader settings

Development of combinatorial contour definitions

Application to the hard-core model beyond lattices

Abstract

Pirogov--Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a combinatorially flexible version of Pirogov--Sinai theory for the hard-core model of independent sets. Our results illustrate that the main conclusions of Pirogov--Sinai theory can be obtained in significantly greater generality than that of $\mathbb Z^{d}$. The main ingredients in our generalization are combinatorial and involve developing appropriate definitions of contours based on the notion of cycle basis connectivity. This is inspired by works of Tim\'{a}r and Georgakopoulos--Panagiotis.