Tightness of the 2.429 upper bound for the Weitz SAW tree connective constant on Z^2

Determine whether the numerical upper bound 2.429 on the connective constant of the Weitz self-avoiding-walk tree T_Weitz(Z^2) under the ordering of Sinclair et al. (2017, Appendix A) is tight; if it is not tight, compute a sharper bound to improve the resulting zero-free threshold beyond λ_c(2.429)=2.538.

Background

To extend zero-freeness results on Z2, the paper leverages the ordering from Sinclair et al. (2017) for the Weitz SAW tree, yielding a numerical upper bound 2.429 on the tree’s connective constant and thus a zero-free regime up to λ_c(2.429)=2.538. This computation-driven bound controls the uniform zero-freeness domain for finite subgraphs of Z2 in their framework.

The authors note the current upper bound is numerical and its tightness is unknown. Improving this bound would directly strengthen the zero-free threshold on Z2, potentially moving closer to the long-standing conjectured value near 3.796.

References

Moreover, the value 2.429 is a numerical upper bound and is not known to be tight; hence the threshold could be improved by sharpening the underlying computation.

— Zero-Freeness of the Hard-Core Model with Bounded Connective Constant  (2604.02746 - Chen et al., 3 Apr 2026) in Appendix: Improved threshold on Z^2 via Weitz trees (Sinclair et al., Appendix A)