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Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions

For the hard-core lattice gas model defined on independent sets weighted by an activity $λ$, we study the critical activity $λ_c(\mathbb{Z}^2)$ for the uniqueness/non-uniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately $3.796$. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree $Δ$ when $λ<λ_c(\mathbb{T}_Δ)$ where $\mathbb{T}_Δ$ is the infinite, regular tree of degree $Δ$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^σ(\mathbb{Z}^2)$ where $σ=(σ_v)_{v\in \mathbb{Z}^2}$ and $σ_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $λ_c(\mathbb{Z}^2)\geqλ_c( \mathbb{T}_4) = 1.675$. Restrepo et al. (2011) improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $λ_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $σ$, SSM does not hold on $T_{\mathrm{saw}}^σ(\mathbb{Z}^2)$ when $λ>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $λ_c(\mathbb{Z}^2)>2.48$.