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Pirogov--Sinai Theory Beyond Lattices

Sarah Cannon, Tyler Helmuth, Will Perkins

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.

Pirogov--Sinai Theory 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 . 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.

Paper Structure

This paper contains 33 sections, 44 theorems, 74 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Hard-Core Model: Background and Main Results
  3. Algorithmic Results
  4. Illustrative Examples
  5. Context and Related Work: Statistical Mechanics
  6. Context and Related Work: Algorithms
  7. Proof Ideas and Outline of Paper
  8. Preliminaries
  9. Graph Notation and Terminology
  10. Ends, Boundaries, and Isoperimetry
  11. Cycle spaces, cycle bases, and basis connectivity
  12. Boundary Conditions for the Hard-Core Model
  13. Finite Graphs and Boundary Conditions
  14. Infinite Graphs
  15. Pirogov--Sinai Theory: Combinatorial Steps
  16. ...and 18 more sections

Key Result

Theorem 1.1

Suppose $G$ is bipartite, vertex transitive with degree $\Delta$, one-ended, and has a $D$-bounded cycle basis $\mathcal{B}$. There exists a $\lambda_{\star}(D,\Delta)<\infty$ such that if $\lambda>\lambda_{\star}(D,\Delta)$ then phase coexistence occurs for the hard-core model on $G$.

Figures (5)

  • Figure 1: A portion of the dice lattice. The vertices in one bipartite class are emphasized.
  • Figure 2: A finite portion of the width-two semi-infinite cylinder graph.
  • Figure 3: A subset of the graph $\mathbb{Z}^2_2$ from Example \ref{['ex:BHW']}.
  • Figure 4: The smallest contour (wavy lines) on $\mathbb{Z}^2$ when $\mathcal{B}$ consists of the length four cycles $(x,x+e_1,x+e_{1}+e_{2},x+e_{2},x)$ for $x\in \mathbb{Z}^{2}$, $e_{1},e_{2}$ the standard unit basis vectors in $\mathbb{R}^{2}$.
  • Figure 5: Left: a hard-core configuration on a subset of $\mathbb{Z}^{2}$. Large circles indicate vertices contained in the hard-core configuration. Dark and light shading indicates even and odd vertices, respectively. Right: the contour $\gamma$ (wavy lines) corresponding to the hard-core configuration on the right if $\mathcal{B}$ is the cycle basis from Figure \ref{['fig:contour']}. The solid grey edges are edges of the interior component $\mathsf{Int}\gamma$. In this example $\mathsf{Int}\gamma=\mathsf{Int}_{\textsf{e}}\gamma$. The contour is an odd contour as the vertices in the exterior component of $\gamma$ that are contained in edges of $\gamma$ are even.

Theorems & Definitions (96)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 1
  • Example 2
  • Example 3
  • ...and 86 more