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Counting independent sets in percolated graphs via the Ising model

Anna Geisler, Mihyun Kang, Michail Sarantis, Ronen Wdowinski

TL;DR

The paper develops a unified framework to count independent sets in percolated graphs by linking $i(G_p)$ to the antiferromagnetic Ising model via the identity $E[Z_{G_p}(oldsymbol{1})]=Z_G(oldsymbol{1},eta)$ at $p=1-e^{-eta}$. It combines Sapozhenko-style graph containers with the cluster expansion and entropy bounds of Peled–Spinka to prove a convergent expansion for the Ising partition function on a broad class of regular bipartite graphs satisfying vertex-isoperimetric properties (Property I). A refined container lemma is established, improving the permissible range of the fugacity $oldsymbol{ u}$ (and temperature) beyond prior results, which in turn yields precise asymptotics for $E[i(G_p)]$ and related quantities in growing-parameter regimes. The results are demonstrated on concrete graph families, including Cartesian products, the middle layer graph, and growing even tori, and yield explicit first-term and higher-order corrections in the expansion. Overall, the work advances both the theoretical understanding of percolated-counts of independent sets and the methodological toolkit for Ising/hard-core models on nontrivial graph families with expansion properties, with potential implications for sampling and approximation algorithms in statistical physics and combinatorics.

Abstract

Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite graphs satisfying certain vertex-isoperimetric properties, extending the work of Kronenberg and Spinka on the percolated hypercube. Combining graph containers with the cluster expansion from statistical physics, we give an expansion of the partition function of the Ising model in certain range of the parameters. Among other applications, we obtain results for even tori of growing side-length. As a tool, we prove a refined container lemma for the Ising model, which mildly improves recent bounds of Jenssen, Malekshahian, and Park.

Counting independent sets in percolated graphs via the Ising model

TL;DR

The paper develops a unified framework to count independent sets in percolated graphs by linking to the antiferromagnetic Ising model via the identity at . It combines Sapozhenko-style graph containers with the cluster expansion and entropy bounds of Peled–Spinka to prove a convergent expansion for the Ising partition function on a broad class of regular bipartite graphs satisfying vertex-isoperimetric properties (Property I). A refined container lemma is established, improving the permissible range of the fugacity (and temperature) beyond prior results, which in turn yields precise asymptotics for and related quantities in growing-parameter regimes. The results are demonstrated on concrete graph families, including Cartesian products, the middle layer graph, and growing even tori, and yield explicit first-term and higher-order corrections in the expansion. Overall, the work advances both the theoretical understanding of percolated-counts of independent sets and the methodological toolkit for Ising/hard-core models on nontrivial graph families with expansion properties, with potential implications for sampling and approximation algorithms in statistical physics and combinatorics.

Abstract

Given a graph , we form a random subgraph by including each edge of independently with probability . We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite graphs satisfying certain vertex-isoperimetric properties, extending the work of Kronenberg and Spinka on the percolated hypercube. Combining graph containers with the cluster expansion from statistical physics, we give an expansion of the partition function of the Ising model in certain range of the parameters. Among other applications, we obtain results for even tori of growing side-length. As a tool, we prove a refined container lemma for the Ising model, which mildly improves recent bounds of Jenssen, Malekshahian, and Park.

Paper Structure

This paper contains 28 sections, 36 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Main themes and results.
  4. Vertex-isoperimetry
  5. Main results
  6. Graph containers and the cluster expansion
  7. Organization
  8. Preliminaries
  9. Notation and basic facts
  10. Abstract polymer models and the cluster expansion
  11. Our polymer model: Proof of \ref{['thm:expansion']}
  12. Graph containers
  13. Bounds from entropy
  14. Bounding weights of polymers with a given approximation
  15. Bounding weights of non-polymers
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $G$ be a $d$-regular $n$-vertex bipartite graph with bipartition $(\mathcal{O}, \mathcal{E})$ satisfying def:property-i. For any fixed constant $\lambda_0>0$ there exists a constant $C_0>0$ such that the following holds. Suppose $\lambda \le \lambda_0$ and $\lambda p \geq \frac{C_0 \log^{3/2} d} where

Theorems & Definitions (60)

  • Definition 1.1: Property I
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Kotecký--Preiss condition kotecky1986cluster
  • Lemma 3.1
  • Lemma 3.1
  • Lemma 3.1
  • ...and 50 more