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<p>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 on bipartite graphs. Our results illustrate that the main conclusions of Pirogov–Sinai theory can be obtained in significantly greater generality than that of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. 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.</p>

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Pirogov–Sinai Theory for the Hard-Core Model Beyond Lattices

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月2日

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  • Sarah Cannon,
  • Tyler Helmuth,
  • Will Perkins

摘要

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 on bipartite graphs. Our results illustrate that the main conclusions of Pirogov–Sinai theory can be obtained in significantly greater generality than that of \(\mathbb {Z}^{d}\) 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.