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

摘要

For the hard-core lattice gas model defined on independent sets weighted by an activity A, we study the critical activity λ_C(Z~2) for the uniqueness threshold on the 2-dimensional integer lattice 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 A when λ < λ_c(T_Δ) where T_Δ is the infinite, regular tree of degree A. His result established a certain decay of correlations property called strong spatial mixing (SSM) on Z~2 by proving that SSM holds on its self-avoiding walk tree T_(saw)(Z~2), and as a consequence he obtained that λ_c(Z~2) > λ_c(T_4) = 1.675. Re-strepo et al. (2011) improved Weitz's approach for the particular case of Z~2 and obtained that λ_c(Z~2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T_(saw)(Z~2) when A > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λ_c(Z~2) > 2.48.