For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λc(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 < λc(TΔ)λ < λc(TΔ) where TΔ 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 Tsaw(ℤ2), and as a consequence he obtained that λc(Z2) ≥ λc(T4) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λc(ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on Tsaw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λc(ℤ2) > 2.48.
|Title of host publication||The Netherlands as an EU Member|
|Subtitle of host publication||Awkward or Loyal Partner?|
|Editors||A. Schout, J. Rood|
|Place of Publication||The Hague / Portland, OR|
|Publisher||Eleven International Publishing|
|Number of pages||312|
|Publication status||Published - 2013|