Pirogov--Sinai Theory Beyond Lattices

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 Timár and Georgakopoulos–Panagiotis.