Analyticity (zero-freeness) threshold on the square lattice up to λ* ≈ 3.796

Establish analyticity of the infinite-volume free energy for the hard-core model on the square lattice Z^2 in a complex neighborhood of (0, λ* ≈ 3.796); equivalently, prove that for every finite induced subgraph H ⊂ Z^2 the independence polynomial Z_H(λ) has no zeros in a complex neighborhood of (0, λ*).

Background

The paper discusses zero-freeness of the hard-core partition function and analyticity of free energy, noting that results based on maximum degree give much smaller thresholds than believed for structured lattices. For Z2, physics and numerical work suggest analyticity up to about 3.796, far beyond the tree threshold λ_c(3)=1.6875.

While rigorous progress has improved lower bounds (e.g., via correlation decay and connective-constant methods up to 2.538), a direct zero-freeness proof achieving the conjectured threshold remains unresolved. This conjecture represents a central open challenge connecting complex zeros and phase transitions on Z2.

References

For instance, on the square lattice $\mathbb{Z}2$, it has been long conjectured and numerically verified that the free energy is analytic (or equivalently, the partition functions of finite subgraphs are zero-free) on a complex neighborhood of $(0, \lambda\ast \approx 3.796)$ .

— Zero-Freeness of the Hard-Core Model with Bounded Connective Constant  (2604.02746 - Chen et al., 3 Apr 2026) in Section 1: Introduction