Coupling with the stationary distribution and improved sampling for colorings and independent sets
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Combinatorial criteria for uniqueness of Gibbs measures
Random Structures & Algorithms
A general lower bound for mixing of single-site dynamics on graphs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Randomly coloring planar graphs with fewer colors than the maximum degree
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Rapid mixing of Gibbs sampling on graphs that are sparse on average
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Phase transition for Glauber dynamics for independent sets on regular trees
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Sampling colourings of the triangular lattice
Random Structures & Algorithms
Dobrushin conditions for systematic scan with block dynamics
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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The paper considers spin systems on the d-dimensional integer lattice ℤd with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we show that if the mixing time of the Glauber dynamics is O(n log n) then spin correlations decay exponentially fast with distance. We also prove the converse implication for monotone systems, and for general systems we prove that exponential decay of correlations implies O(n log n) mixing time of a dynamics that updates sufficiently large blocks (rather than single sites). While the above equivalence was already known to hold in various forms, we give proofs that are purely combinatorial and avoid the functional analysis machinery employed in previous proofs. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004Supported by EPSRC grant “Sharper Analysis of Randomised Algorithms: a Computational Approach” and by EC IST Project RAND-APX.Supported in part by NSF grants CCR-9820951 and CCR-0121555, and by DARPA Cooperative Agreement F30602-00-2-060.