Improved mixing condition on the grid for counting and sampling independent sets
 Ricardo Restrepo,
 Jinwoo Shin,
 Prasad Tetali,
 Eric Vigoda,
 Linji Yang
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The hardcore model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set I in a graph G is weighted proportionally to λ^{I}, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ ≥ 3, is a well known computationally challenging problem. More concretely, let \({\lambda_c(\mathbb{T}_\Delta)}\) denote the critical value for the socalled uniqueness threshold of the hardcore model on the infinite Δregular tree; recent breakthrough results of Weitz (Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149, 2006) and Sly (Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296, 2010) have identified \({\lambda_c(\mathbb{T}_\Delta)}\) as a threshold where the hardness of estimating the above partition function undergoes a computational transition. We focus on the wellstudied particular case of the square lattice \({\mathbb{Z}^2}\) , and provide a new lower bound for the uniqueness threshold, in particular taking it well above \({\lambda_c(\mathbb{T}_4)}\) . Our technique refines and builds on the tree of selfavoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hardcore model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to \({\mathbb{Z}^2}\) we prove that strong spatial mixing holds for all λ < 2.3882, improving upon the work of Weitz that held for λ < 27/16 = 1.6875. Our results imply a fullypolynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hardcore distribution.
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 Title
 Improved mixing condition on the grid for counting and sampling independent sets
 Journal

Probability Theory and Related Fields
Volume 156, Issue 12 , pp 7599
 Cover Date
 20130601
 DOI
 10.1007/s0044001204218
 Print ISSN
 01788051
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Lattice gas
 Gibbs measures
 Phase transition
 Approximation algorithm
 Glauber dynamics
 82B20
 68Q25
 60J10
 Industry Sectors
 Authors

 Ricardo Restrepo ^{(1)}
 Jinwoo Shin ^{(2)}
 Prasad Tetali ^{(3)}
 Eric Vigoda ^{(2)}
 Linji Yang ^{(2)}
 Author Affiliations

 1. Universidad de Antioquia, Medellin, Colombia
 2. School of Computer Science, Georgia Institute of Technology, Atlanta, GA, 30332, USA
 3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA