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arxiv: 2604.02746 · v1 · submitted 2026-04-03 · ❄️ cond-mat.stat-mech · cs.DS· math-ph· math.MP· math.PR

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

Yuan Chen , Shuai Shao , Ke Shi This is my paper

Pith reviewed 2026-05-13 18:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.DSmath-phmath.MPmath.PR
keywords hard-core modelzero-freenessconnective constantpartition functionanalyticityfree energycorrelation decayself-avoiding walks
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The pith

The hard-core partition function stays zero-free in a complex neighborhood of the real interval up to the connective-constant threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that graphs whose self-avoiding-walk growth is bounded by a lower connective constant μ have hard-core partition functions without zeros in a strip around the positive real axis for activities below the tree uniqueness threshold λ_c(μ). This replaces the classical maximum-degree bound with a tighter structural measure that better captures lattices and other sparse graphs. As a direct consequence the free energy density on infinite lattices becomes analytic and unique up to the same connective-constant threshold. Readers care because the result widens the region where correlation-decay methods and uniqueness statements are guaranteed to hold.

Core claim

For any graph family with lower connective constant μ the hard-core partition function is zero-free in a complex neighborhood of the interval [0, λ] for all λ < λ_c(μ). This establishes uniqueness and analyticity of the free energy density for infinite lattices up to the connective-constant threshold.

What carries the argument

Block contraction technique that lifts real-line correlation decay to a complex strip without introducing zeros, using a lower-bound definition of connective constant via k-depth self-avoiding walks.

If this is right

  • Free energy density is analytic and unique for infinite lattices whose connective constant is bounded by μ, for all activities below λ_c(μ).
  • Correlation-decay algorithms remain valid for complex activities inside the zero-free region.
  • Uniqueness of the Gibbs measure holds up to the connective-constant threshold on the same graph families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same connective-constant parameter may govern zero-freeness for other polymer models or two-spin systems.
  • On regular lattices the new threshold is strictly larger than the degree-based one, potentially tightening known phase-transition bounds.
  • The block-contraction method could be tested on finite approximations to lattices to locate the first zero numerically.

Load-bearing premise

The block contraction lifts real correlation decay to a complex strip without creating partition-function zeros.

What would settle it

Exhibit a finite graph with lower connective constant μ and an activity λ below λ_c(μ) whose hard-core partition function has a zero inside a fixed complex neighborhood of the real interval.

read the original 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 $\lambda_c(\Delta-1)$ determined by the maximum degree $\Delta$. However, for many graph classes, such as regular lattices, the connective constant $\sigma$ 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 $\lambda_c(\sigma)$, 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 $\mu$, the partition function is zero-free in a complex neighborhood of the interval $[0, \lambda]$ for all $\lambda < \lambda_c(\mu)$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for any graph family with a lower connective constant μ, the hard-core partition function is zero-free in a complex neighborhood of [0, λ] for all λ < λ_c(μ). It introduces a definition of the connective constant for finite graphs via a lower bound on k-depth self-avoiding walks and employs a block contraction technique to lift real-line correlation decay to a complex strip. This yields uniqueness and analyticity of the free energy density on infinite lattices up to the connective-constant threshold, extending prior maximum-degree results.

Significance. If the central lifting argument holds, the result meaningfully strengthens zero-freeness bounds by replacing maximum-degree thresholds with the tighter connective-constant measure, which is especially relevant for lattices and other structured graphs. The finite-graph definition of the connective constant is a useful technical contribution that could support future algorithmic work on approximation and sampling.

major comments (2)
  1. [§4] §4 (Block contraction argument): the manuscript asserts that the block contraction lifts real correlation decay to a complex neighborhood without introducing zeros, yet supplies no explicit contraction factor, strip width, or uniform bound derived from μ. Without these quantities it is impossible to confirm that the neighborhood remains non-empty for λ approaching λ_c(μ) from below.
  2. [§2] §2 (Definition of lower connective constant): the lower-bound definition via k-depth SAW counts controls real branching, but the text does not demonstrate that this bound remains sufficient to dominate effective branching under complex weights inside the claimed strip; a counter-example or explicit majorization argument is needed.
minor comments (2)
  1. [Abstract] The abstract refers to 'a proper definition' of the connective constant; the main text should state the definition in a numbered display equation for easy reference.
  2. [References] Ensure the reference list includes all prior correlation-decay results that are invoked for the real-line case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will revise the manuscript to improve clarity on the explicit bounds and majorization.

read point-by-point responses

  1. Referee: [§4] §4 (Block contraction argument): the manuscript asserts that the block contraction lifts real correlation decay to a complex neighborhood without introducing zeros, yet supplies no explicit contraction factor, strip width, or uniform bound derived from μ. Without these quantities it is impossible to confirm that the neighborhood remains non-empty for λ approaching λ_c(μ) from below.

    Authors: We agree that the presentation in §4 would benefit from explicit quantities. The block contraction argument derives a contraction factor strictly less than 1 from the real-line decay rate (itself controlled by μ via the connective constant threshold), which in turn determines a positive strip width and uniform bound on the zero-free neighborhood. In the revision we will state these explicit expressions (including their dependence on μ and λ) and verify that they remain positive for all λ < λ_c(μ), confirming the neighborhood is non-empty. revision: yes

  2. Referee: [§2] §2 (Definition of lower connective constant): the lower-bound definition via k-depth SAW counts controls real branching, but the text does not demonstrate that this bound remains sufficient to dominate effective branching under complex weights inside the claimed strip; a counter-example or explicit majorization argument is needed.

    Authors: The lower connective constant is defined precisely so that the k-depth SAW lower bound dominates the branching factor. We will add a short majorization lemma in the revised §2 showing that, inside the complex strip produced by the block contraction, the modulus of the effective branching is at most the real branching factor controlled by the given SAW count lower bound. This establishes the required domination without counter-examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces an independent definition of the lower connective constant μ for finite graphs via a lower bound on k-depth self-avoiding walk counts, then applies a block contraction technique to lift established real-line correlation decay (from prior results) to a complex zero-free neighborhood for λ < λ_c(μ). No step reduces by construction to its own inputs, fitted parameters renamed as predictions, or load-bearing self-citation chains; the central zero-freeness claim and free-energy analyticity consequence remain independent of the target result itself. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of self-avoiding walks plus a new block-contraction argument introduced to extend real correlation decay into the complex plane.

axioms (2)
  • domain assumption Lower connective constant μ is well-defined for finite graphs via a lower bound on the number of k-depth self-avoiding walks
    Definition introduced in the paper and used to set the threshold λ_c(μ)
  • ad hoc to paper Block contraction lifts real-line correlation decay to a complex neighborhood without introducing zeros
    Core technical step of the proof; no external reference given in abstract

pith-pipeline@v0.9.0 · 5570 in / 1391 out tokens · 66399 ms · 2026-05-13T18:39:56.751875+00:00 · methodology

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Reference graph

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