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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Vera, Juan C. Vigoda, Eric Yang, Linji
Copyright Year	2015
Abstract	For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz [Proceedings of the $38$th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 140--149]. Weitz presented a fully polynomial-time approximation scheme for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$, where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$, where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. [Probab. Theory Related Fields, 156 (2013), pp. 75--99] improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.
Starting Page	1895
Ending Page	1915
Page Count	21
File Format	PDF
ISSN	08954801
DOI	10.1137/140976923
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-10-20
Access Restriction	Subscribed
Subject Keyword	strong spatial mixing Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs approximate counting branching matrices MCMC Markov chains Linear programming linear programming
Content Type	Text
Resource Type	Article
Subject	Mathematics

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