Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On expectations and variances in the hard-core model

Weiyuan Zhang, Kexiang Xu

Abstract

The hard-core model can be used to understand the numbers of independent sets in graphs in extremal graph theory. The occupancy fraction, defined as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in hard-core model. Davies \textit{et al.} (2017) established an upper bound on the occupancy fraction for $d$-regular graphs, and Perarnau and Perkins (2018) derived a corresponding bound on it for graphs with given girth. Inspired by their work, we provide the tight upper and lower bounds on occupancy fraction in $n$-vertex graphs with independence number $α$, extending the classical results on bounds for independence polynomials. We also prove a relevant conjecture posed by Davies \textit{et al.} (2025) to this topic.

On expectations and variances in the hard-core model

Abstract

The hard-core model can be used to understand the numbers of independent sets in graphs in extremal graph theory. The occupancy fraction, defined as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in hard-core model. Davies \textit{et al.} (2017) established an upper bound on the occupancy fraction for -regular graphs, and Perarnau and Perkins (2018) derived a corresponding bound on it for graphs with given girth. Inspired by their work, we provide the tight upper and lower bounds on occupancy fraction in -vertex graphs with independence number , extending the classical results on bounds for independence polynomials. We also prove a relevant conjecture posed by Davies \textit{et al.} (2025) to this topic.

Paper Structure

This paper contains 8 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Proofs on occupancy fraction
  3. Proof of Theorem \ref{['upper_bound']}
  4. Proof of Theorem \ref{['lowerbound']}
  5. Proof of Corollary \ref{['PG_bound']}
  6. Proofs on Variance fraction
  7. Proof of Theorem \ref{['th:kn']}
  8. Proof of Theorem \ref{['th:kdelta']}

Key Result

Theorem 1.2

Let $G\in \mathcal{G}(n,\alpha)$. Then, for any $\lambda> 0$, we have where $k$ is the number of components of $Z(n,\alpha)$ of order $\lfloor\frac{n}{\alpha}\rfloor$ with $0< k\leq \alpha$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (25)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 15 more