SIAM Journal on Computing
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Approximately counting up to four (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Simulated Annealing in Convex Bodies and an 0*(n4) Volume Algorithm
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Counting and sampling H-colourings
Information and Computation
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Randomly Coloring Constant Degree Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A second threshold for the hard-core model on a Bethe lattice
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Combinatorial criteria for uniqueness of Gibbs measures
Random Structures & Algorithms
Random Structures & Algorithms
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems
SIAM Journal on Computing
A deterministic approximation algorithm for computing the permanent of a 0,1 matrix
Journal of Computer and System Sciences
Approximating partition functions of the two-state spin system
Information Processing Letters
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Correlation decay and deterministic FPTAS for counting colorings of a graph
Journal of Discrete Algorithms
Left and right convergence of graphs with bounded degree
Random Structures & Algorithms
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In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494…)n-many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximately ${[q{(1 - {1 \over q})}^{r \over 2} ]^n}$ n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Preliminary version of this paper appeared in Proceedings of the Seventeenth ACM-SIAM Symposium on Discrete Algorithms 2006. ArXive version.