The paradox of The Netherlands: Why a successful economy is struggling?

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 λ < λ_c(T_Δ)λ < λ_c(T_Δ) where 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 ℤ² by proving that SSM holds on its self-avoiding walk tree T_saw(ℤ²), and as a consequence he obtained that λ_c(Z²) ≥ λ_c(T₄) = 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.