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On the critical fugacity of the hard-core model on regular bipartite graphs

Daniel Hadas, Ron Peled

Abstract

We establish long-range order for the hard-core model on a finite, regular bipartite graph above a threshold fugacity given in terms of expansion parameters of the graph. The result applies to the $d$-dimensional hypercube graph and, more generally, to $d$-dimensional discrete tori of fixed side length, proving long-range order at fugacities $λ\geΩ(\frac{\log d}{d})$. Furthermore, we use reflection positivity to transfer the result to the lattice $\mathbb{Z}^{d}$, verifying the long-standing belief that its critical fugacity is of the form $d^{-1+o(1)}$ as $d\to\infty$.

On the critical fugacity of the hard-core model on regular bipartite graphs

Abstract

We establish long-range order for the hard-core model on a finite, regular bipartite graph above a threshold fugacity given in terms of expansion parameters of the graph. The result applies to the -dimensional hypercube graph and, more generally, to -dimensional discrete tori of fixed side length, proving long-range order at fugacities . Furthermore, we use reflection positivity to transfer the result to the lattice , verifying the long-standing belief that its critical fugacity is of the form as .

Paper Structure

This paper contains 45 sections, 24 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. The hard-core model
  3. The hard-core model on $\mathbb{Z}^{d}$
  4. The hard-core model on discrete tori and other regular bipartite graphs
  5. Discrete tori
  6. Regular bipartite graphs
  7. Outline
  8. Notation
  9. Entropy and the free energy functional
  10. Shannon entropy
  11. The free energy functional
  12. The free energy functional of the hard-core model
  13. Long-range order for regular bipartite finite graphs
  14. Coarse graining and global expansion
  15. Using the global expansion
  16. ...and 30 more sections

Key Result

Theorem 1.1

There exists $C>0$ such that the hard-core model on $\mathbb{Z}^{d}$ admits multiple Gibbs measures at each fugacity $\lambda>C\frac{\log d}{d}$ in dimensions $d\ge2$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 1.5
  • Lemma 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9: Sharpness
  • Lemma 1.10
  • ...and 44 more