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

2013/06/03 by Juan C. Vera, Vera, Juan C., Eric Vigoda +3
Mathematics · Physics and Astronomy · #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1306.0431

openalex publication_date 2013/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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/non-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 λ2.388. In this paper, we establish an upper bound for this approach, by showing that, for all σ, SSM does not hold on T_saw^σ(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.

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