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

Abstract

For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λ c (ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. 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 \(\lambda<\lambda_c({\mathbb T}_\Delta)\) where \({\mathbb T}_\Delta\) is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ2 by proving that SSM holds on its self-avoiding walk tree T saw(ℤ2), and as a consequence he obtained that \(\lambda_c({\mathbb Z}^2)\geq\lambda_c( {\mathbb T}_4) = 1.675\). Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λ c (ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T saw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λ c (ℤ2) > 2.48.KeywordsHard-core ModelUniquenessPhase TransitionStrong Spatial MixingApproximate Counting