Zero-Freeness of the Hard-Core Model with Bounded Connective Constant

Abstract: We study the zero-free regions of the partition function of the hard-core model on finite graphs and their implications for the analyticity of the free energy on infinite lattices. Classically, zero-freeness results have been established up to the tree uniqueness threshold $λ_c(Δ-1)$ determined by the maximum degree $Δ$. However, for many graph classes, such as regular lattices, the connective constant $σ$ provides a more precise measure of structural complexity than the maximum degree. While recent approximation algorithms based on correlation decay and Markov chain Monte Carlo have successfully exploited the connective constant to improve the threshold to $λ_c(σ)$, analogous results for complex zero-freeness have been lacking. In this paper, we bridge this gap by introducing a proper definition of the connective constant for finite graphs based on a lower bound on the number of $k$-depth self-avoiding walks. We prove that for any graph family with a lower connective constant $μ$, the partition function is zero-free in a complex neighborhood of the interval $[0, λ]$ for all $λ< λ_c(μ)$. As a direct consequence, we establish the uniqueness and analyticity of the free energy density for infinite lattices up to the connective constant threshold, extending the known regions derived from maximum degree bounds. Our proof utilizes a block contraction technique that lifts the correlation decay property from a real interval to a strip-like complex neighborhood.

• The paper establishes zero-free regions for the hard-core model's partition function up to the connective constant threshold.
• It introduces a novel k-depth lower connective constant and employs block contraction with potential function analysis.
• This result enhances deterministic approximation algorithms and yields improved analytic insights into lattice free energy.

Zero-Freeness of the Hard-Core Model with Bounded Connective Constant

Introduction and Motivation

The study addresses the zero-free regions for the partition function of the hard-core model (i.e., independence polynomial) on families of finite graphs parameterized by the connective constant and explores the analytic properties of the associated free energy in the thermodynamic limit. Historically, the zero-freeness of the partition function—a property essential in both statistical physics and the design of deterministic algorithms—was characterized primarily by the maximum degree of the underlying graph, with thresholds related to the uniqueness condition on the infinite tree. However, these degree-based thresholds are suboptimal for structured infinite graphs, such as regular lattices, where the combinatorial complexity is more precisely captured by the connective constant, which reflects the growth rate of self-avoiding walks (SAWs).

While prior algorithmic work established improved thresholds for correlation decay and sampling using the connective constant [sinclair2017spatial, efthymiou2026sampling], analogous progress on zero-freeness—essential for ensuring the analyticity of the free energy and for Barvinok-style deterministic approximation algorithms—has lagged. This paper resolves this gap by establishing zero-free regions for the hard-core partition function up to the threshold determined by the connective constant of the underlying graph family, providing both new structural insights and algorithmic implications.

Main Theoretical Contributions

Definition of Connective Constant for Finite Graphs

To generalize the notion of connective constant to finite graphs in a manner suitable for controlling zero-free regions, the authors introduce the "lower connective constant" $\mu_{\inf}$, defined as the infimum over $k$ of the $k$-depth connective constants:

$$\mu_{k}(G) = \sup_{v \in V} \left(\mathcal{N}_{\leq k}(G, v)\right)^{1/k}, \quad \mu_{\inf}(\mathcal{H}) = \inf_{k\geq 1} \mu_{k}(\mathcal{H}),$$

where $\mathcal{N}_{\leq k}(G,v)$ is the number of SAWs of length at most $k$ from $v$. This notion aligns with the standard connective constant for infinite transitive graphs and remedies shortcomings in existing log-depth definitions, which fail to guarantee uniform zero-freeness.

Main Zero-Freeness Theorem

The principal result establishes that for any family $\mathcal{H}$ of finite graphs with bounded lower connective constant $\mu_{\inf}(\mathcal{H})$, and for any fugacity $\lambda < \lambda_c(\mu_{\inf}(\mathcal{H}))$ (where $k$0 is the uniqueness threshold for the infinite $k$1-regular tree), there exists a strip-like complex neighborhood of $k$2 where the partition function $k$3 is zero-free for all $k$4. This result lifts degree-based zero-freeness conditions to the strictly finer setting of bounded connective constant.

A direct corollary is the analyticity and uniqueness of the free energy density for infinite lattices up to this improved threshold, extending the known Lee–Yang region as traditionally derived from maximum degree arguments to thresholds determined by the lattice's connective constant.

Notably, for lattices such as $k$5, this raises the rigorous analyticity threshold to $k$6 and, with Weitz-tree refinements, up to $k$7, significantly beyond the traditional tree threshold $k$8.

Technical Approach: Block Contraction and Analytic Continuation

The proof adapts and strengthens real-to-complex contraction methodology from spatial mixing analyses. Key technical innovations include:

• Block contraction: Instead of analyzing the classical one-step recurrence (root-to-children), the proof extends contraction to $k$9-step blocks, which controls error propagation across large interfaces and permits bounding in terms of the $k$0-depth connective constant rather than maximum degree.
• Potential function analysis: An explicit analytic mapping (e.g., $k$1) is used to establish contraction in an appropriate norm, adapting the strong spatial mixing framework [sinclair2017spatial] and strengthening it for zero-freeness.
• Complex extension: The real contraction property is analytically continued to complex neighborhoods by invoking a real-to-complex extension theorem [shao2021contraction], ensuring that small perturbations in the fugacity do not induce zeros.

This block-contraction framework guarantees that for any $k$2 in the strip neighborhood and any rooted SAW tree with $k$3-depth connective constant $k$4, the root occupation ratio avoids the pole $k$5 and thus ensures nonvanishing partition functions.

Implications for Infinite Lattices

For infinite, quasi-transitive graphs, including regular lattices, the correspondence between the lower connective constant for finite subgraphs and the standard connective constant for the infinite graph is established. Under mild periodicity/homogeneity assumptions, analyticity of the limiting free energy in a zero-free neighborhood follows via Vitali's theorem and uniform zero-freeness for the exhaustion sequence.

Numerical and Structural Results

The paper highlights that for several classical lattices, the connective constant threshold is strictly larger than the one implied by maximum degree. For instance, on the square lattice $k$6, the new analytic regime at $k$7 (or $k$8 using Weitz-tree bounds) greatly exceeds the uniqueness threshold for degree $k$9 graphs. For more exotic lattices, the gap is even more pronounced.

A critical theoretical claim is the demonstration that log-depth-based connective constant bounds do not suffice for zero-freeness, by constructing explicit families (e.g., finite regular trees) where zeros accumulate at points inside the degree-based uniqueness region, thus necessitating the $$\mu_{k}(G) = \sup_{v \in V} \left(\mathcal{N}_{\leq k}(G, v)\right)^{1/k}, \quad \mu_{\inf}(\mathcal{H}) = \inf_{k\geq 1} \mu_{k}(\mathcal{H}),$$0-depth definition.

Algorithmic and Physical Implications

Zero-freeness of the partition function is central to deterministic approximation algorithms for evaluation and sampling in the hard-core model, specifically via Barvinok's method and related complex-analytic techniques [barvinok2016combinatorics, patel2017deterministic]. By expanding the zero-free region beyond degree-based boundaries, the results potentially enable FPTASes for larger regimes of fugacity in graphs with bounded connective constant, including regular lattices pertinent to statistical physics and combinatorial enumeration.

From a physical perspective, the analyticity of the free energy in the complex fugacity plane characterizes the absence of phase transitions and connects directly to the Lee–Yang program for understanding phase structure. The expanded analytic region sharpens theoretical predictions for critical behavior in lattice systems and informs thresholds for computational intractability.

Conclusion

This work rigorously establishes that the zero-freeness of the hard-core partition function, and consequently the analyticity and uniqueness of the infinite-volume free energy, can be characterized in terms of the connective constant rather than the traditional maximum degree. The result closes a notable gap between algorithmic thresholds (correlation decay, MCMC) and zero-freeness, and extends the reach of analytic and algorithmic results into domains previously accessible only via numerical or nonrigorous methods. The methodology of block contraction and its real-to-complex extension provides a general framework potentially adaptable to other spin systems and graphical models, inviting further advances in algorithmic statistical mechanics and structural graph theory.

References

• [sinclair2017spatial]: Optimal correlation decay and spatial mixing using connective constant.
• [efthymiou2026sampling]: Optimal sampling using local connective constant for two-spin systems.
• [barvinok2016combinatorics], [patel2017deterministic]: Barvinok’s interpolation and deterministic algorithms.
• [shao2021contraction]: Contraction framework for zero-freeness and spatial mixing.

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