Pirogov--Sinai Theory Beyond Lattices

2024/11/12 by Sarah Cannon, Cannon, Sarah, Tyler Helmuth +3 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · #Advanced Algebra and Logic #Cellular Automata and Applications #DNA and Biological Computing #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2411.07809

openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Timár and Georgakopoulos--Panagiotis.

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• Extended regime of nematic order in an interacting monomer-dimer model of Heilmann and Lieb
• The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence

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