Paper Info
Title
Improved Bounds on the Phase Transition for the Hard-Core Model in 2 Dimensions.
Abstract
For the hard-core lattice gas model defined on independent sets weighted by an activity lambda, we study the critical activity lambda(c)(Z(2)) for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice 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 38th 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)(T-Delta), where 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 Z(2) by proving that SSM holds on its self-avoiding walk tree T-saw(sigma)(Z(2)), where sigma = (sigma(v)) v is an element of Z(2) and sigma(v) is an ordering on the neighbors of vertex v. As a consequence he obtained that lambda(c)(Z(2)) >= lambda(c)(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 Z(2) and obtained that lambda(c)(Z(2)) > 2.388. In this paper, we establish an upper bound for this approach, by showing that, for all s, SSM does not hold on T-saw(sigma)(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)(Z(2)) > 2.48.
Year
DOI
Venue
2015
10.1137/140976923
SIAM JOURNAL ON DISCRETE MATHEMATICS
Keywords
DocType
Volume
approximate counting,MCMC,strong spatial mixing,branching matrices,linear programming
Journal
29
Issue
ISSN
Citations 
4
0895-4801
6
PageRank 
References 
Authors
0.50
7
3
Name
Order
Citations
PageRank
Juan Carlos Vera1293.33
Eric Vigoda274776.55
Linji Yang3585.17