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 $d$-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.

• The paper establishes sharp extremal bounds for the occupancy fraction E_G(λ), showing the maximum is attained by the complement of a Turán graph and the minimum by Kₙ₋ₐ ∨ (α K₁) for small λ.
• It derives matching upper and lower bounds for the independence polynomial P_G(λ), thereby generalizing classical Turán-type inequalities to all positive λ.
• It resolves a conjecture by Davies et al., proving the minimal variance fraction and precisely characterizing the extremal graphs governing both expectation and variance metrics.

Expectations and Variances in the Hard-Core Model: Sharp Extremal Bounds

Introduction

The hard-core model is a fundamental probability distribution on independent sets of a graph, parameterized by an activity $\lambda > 0$, assigning to each independent set $I$ a probability proportional to $\lambda^{|I|}$. The occupancy fraction, $E_G(\lambda)$, formalized as the logarithmic derivative of the independence (partition) polynomial, encapsulates the expected relative size of independent sets under this model. The variance fraction $V_G(\lambda)$ similarly encodes the rescaled variance. These quantities are deeply linked to broader extremal and probabilistic questions in graph theory, and have motivated a substantial body of work seeking tight bounds under various structural constraints.

This paper significantly advances the understanding of extremal properties of $E_G(\lambda)$ and $V_G(\lambda)$ among $n$-vertex graphs with fixed independence number $\alpha$, bridging and generalizing several classical and modern results. It settles a recent conjecture posed in [DST2025] regarding the minimum possible variance fraction, and precisely characterizes the extremal graphs for both expectation and variance metrics.

Main Contributions

1. Sharp Bounds on Occupancy Fraction for $(n, \alpha)$-Graphs

The authors prove that among all graphs with $I$0 vertices and independence number $I$1, the maximum occupancy fraction $I$2 for any $I$3 is achieved when $I$4 is the complement of the Turán graph $I$5, referred to as $I$6. Explicitly:

$I$7

where $I$8 is determined by the partition of $I$9 into nearly equal sizes.

The lower bound is established for small $\lambda^{|I|}$0 ($\lambda^{|I|}$1), with the extremal graph $\lambda^{|I|}$2, yielding

$\lambda^{|I|}$3

These bounds generalize classical Zykov-type inequalities (on counts of independent sets for fixed independence number), extending them from integer solutions ($\lambda^{|I|}$4) to all positive $\lambda^{|I|}$5, and give tight control over the expected size of random independent sets in this rigorous general context.

2. Tight Bounds on the Independence Polynomial

From these occupancy fraction results, matching upper and lower bounds are derived for the independence polynomial itself:

$\lambda^{|I|}$6

The upper bound extends previous generalized Turán-type bounds to all $\lambda^{|I|}$7, resolving a long-standing gap in the hierarchy of extremal independence polynomial inequalities.

3. Resolution of the Variance Conjecture

The paper proves the conjecture of Davies et al. [DST2025] regarding the minimal variance fraction:

• For any $\lambda^{|I|}$8-vertex graph $\lambda^{|I|}$9 and all $E_G(\lambda)$0,

$E_G(\lambda)$1

• For graphs with maximum degree $E_G(\lambda)$2,

$E_G(\lambda)$3

Equality is achieved for complete graphs with the respective orders.

These results are obtained via sophisticated combinatorial and probabilistic techniques, including careful symmetrization arguments, Karamata’s inequality, and law of total variance decompositions.

Theoretical and Practical Implications

The explicit determination of extremal graphs and sharp analytic expressions for $E_G(\lambda)$4 and $E_G(\lambda)$5 fundamentally enhances the understanding of the hard-core model's behavior under structural constraints. These results directly unify and extend various strands of extremal combinatorics:

• Graph containers and probabilistic methods for counting and sampling independent sets.
• Entropy and correlation inequalities in statistical physics and information theory.
• Applications to Ramsey theory and bounds for Ramsey numbers, as the occupancy and variance properties control the feasibility and likelihood of large independent sets in sparse graphs.

The verification of the minimum variance conjecture further provides tools for analyzing concentration in sampling algorithms and the limiting behavior of random independent sets, with implications for approximate counting (FPRAS), coloring, and hardness of approximation.

Methodological Highlights

The proof framework leverages:

• Majorization and Karamata’s inequality to maximize the sum $E_G(\lambda)$6 over partitions, identifying Turán graphs as optimizers for occupancy.
• Detailed probabilistic decompositions for variance, including the law of total variance and covariance bounds, ensuring sharp control over second moments.
• Symmetrization transformations to reduce to multipartite (Turán) structures, a classical yet powerful approach for extremal problems.

The arguments are robust and admit further generalization, potentially to other models (matching polynomials, weighted homomorphisms) or to broader classes of graph parameters.

Prospects for Future Research

The results invite several lines of ongoing and future inquiry:

• Determining the threshold $E_G(\lambda)$7 (see the authors' conjecture) for uniqueness of minimizing graphs in the lower occupancy bound, as well as critical phenomena near this threshold.
• Extending the methods to hypergraphs, or to other hard constraints (e.g., forbidden induced subgraphs).
• Utilizing these sharp moment bounds to analyze mixing and convergence for Markov chains (Glauber dynamics, etc.) and to develop new deterministic or randomized algorithms for combinatorial optimization problems.
• Investigating corresponding tight inequalities for related models, such as the monomer-dimer and Potts models, and the interplay with graph entropy frameworks.

Conclusion

This paper provides definitive extremal bounds for expectations and variances in the hard-core model for graphs with prescribed independence number, settling significant open problems in the field. The identification of Turán and specific join graphs as extremal examples unifies classical and modern combinatorial extremal theory, and the methods developed here are likely to have lasting influence on the analysis of partition functions, concentration phenomena, and probabilistic combinatorics more broadly (2604.01717).