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

  • Juan C. Vera
  • Eric Vigoda
  • Linji Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8096)

Abstract

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 \(\lambda<\lambda_c({\mathbb T}_\Delta)\) 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 Tsaw(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 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.

Keywords

Hard-core Model Uniqueness Phase Transition Strong Spatial Mixing Approximate Counting 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Juan C. Vera
    • 1
  • Eric Vigoda
    • 2
  • Linji Yang
    • 2
  1. 1.Department of Econometrics and Operations ResearchTilburg UniversityTilburgThe Netherlands
  2. 2.School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA

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