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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Abstract
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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 Alm S.E.: Upper bounds for the connective constant of selfavoiding walks. Combin. Probab. Comput. 4(2), 115–136 (1993)
 Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput. 865–877 (1984)
 Baxter R.J., Enting I.G., Tsang S.K.: Hardsquare lattice gas. J. Stat. Phys. 22(4), 465–489 (1980) CrossRef
 Beffara, V., DuminilCopin, H.: The selfdual point of the twodimensional random cluster model is critical for q ≥ 1. Probab. Theory Relat. Fields. Preprint available from the arXiv at: http://arxiv.org/abs/1006.5073
 van den Berg J., Ermakov A.: A new lower bound for the critical probability of site percolation on the square lattice. Random Struct. Algorithms 8(3), 199–212 (1996) CrossRef
 van den Berg J., Steif J.E.: Percolation and the hardcore lattice gas model. Stochastic Processes Appl. 49(2), 179–197 (1994) CrossRef
 Brightwell G.R., Häggström O., Winkler P.: Nonmonotonic behavior in hardcore and Widom–Rowlinson models. J. Stat. Phys. 94(3), 415–435 (1999) CrossRef
 Cesi F.: Quasi–factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Relat. Fields 120(4), 569–584 (2001) CrossRef
 Dobrushin R.L.: The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct. Anal. Appl. 2(4), 302–312 (1968) CrossRef
 Dobrushin R.L., Shlosman S.B.: Constructive unicity criterion. In: Fritz, J., Jaffe, A., Szasz, D. (eds) Statistical Mechanics and Dynamical Systems, pp. 347–370. Birkhäuser, New York (1985)
 Dyer M.E., Sinclair A., Vigoda E., Weitz D.: Mixing in time and space for lattice spin systems: a combinatorial view. Random Struct. Algorithms 24(4), 461–479 (2004) CrossRef
 Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hardcore model. In: Proocedings of the 15th International Workshop, RANDOM, pp. 567–578 (2011)
 Gaunt D.S., Fisher M.E.: Hardsphere lattice gases. I. Planesquare lattice. J. Chem. Phys. 43(8), 2840–2863 (1965) CrossRef
 Georgii H.O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988) CrossRef
 Goldberg L., Martin R., Paterson M.: Strong spatial mixing for lattice graphs with fewer colours. SIAM J. Comput. 35(2), 486–517 (2005) CrossRef
 Greenhill C.: The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complex. 9(1), 52–72 (2000) CrossRef
 Jerrum M.R., Valiant L.G., Vazirani V.V.: Random generation of combinatorial structures from a uniform distribution distribution. Theor. Comput. Sci. 43(23), 169–186 (1986) CrossRef
 Kelly F.P.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991) CrossRef
 Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)
 Lyons R.: The Ising model and percolation on trees and treelike graphs. Commun. Math. Phys. 125(2), 337–353 (1989) CrossRef
 Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics (SaintFlour, 1997), Lecture Notes in Mathematics, vol. 1717, pp. 93–191 (1998)
 Martinelli F., Olivieri E.: Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Commun. Math. Phys. 161(3), 447–486 (1994) CrossRef
 Martinelli F., Olivieri E.: Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Commun. Math. Phys. 161(3), 487–514 (1994) CrossRef
 Onsager L.: Crystal statistics. I. A twodimensional model with an order–disorder transition. Phys. Rev. Lett. 65(3–4), 117–149 (1944)
 Pönitz A., Tittman P.: Improved upper bounds for selfavoiding walks in ${\mathbb{Z}^d}$ . Electr. J. Combin. 7(1), R21 (2000)
 Rácz Z.: Phase boundary of Ising antiferromagnets near H = H _{ c } and T = 0: results from hardcore lattice gas calculations. Phys. Rev. B 21(9), 4012–4016 (1980) CrossRef
 Radulescu, D.C.: A computerassisted proof of uniqueness of phase for the hardsquare lattice gas model in two dimensions. PhD dissertation, Rutgers University, New Brunswick, NJ, USA (1997)
 Radulescu D.C., Styer D.F.: The Dobrushin–Shlosman phase uniqueness criterion and applications to hard squares. J. Stat. Phys. 49(1–2), 281–295 (1987) CrossRef
 Randall, D.: Slow mixing of Glauber dynamics via topological obstructions. In: Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 870–879 (2006)
 Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296 (2010)
 Štefankovič D., Vempala S., Vigoda E.: Adaptive simulated annealing: a nearoptimal connection between sampling and counting. J. ACM 56(3), 1–36 (2009)
 Tarski A.: A decision method for elementary algebra and geometry, 2nd edn. University of California Press, California (1951)
 Valiant L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979) CrossRef
 Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149 (2006)
 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