Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
Fast convergence of the Glauber dynamics for sampling independent sets
Random Structures & Algorithms - Special issue on statistical physics methods in discrete probability, combinatorics, and theoretical computer science
Torpid Mixing of Some Monte Carlo Markov Chain Algorithms in Statistical Physics
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$
Combinatorics, Probability and Computing
Slow Mixing of Markov Chains Using Fault Lines and Fat Contours
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Convergence rates of Markov chains for some self-assembly and non-saturated Ising models
Theoretical Computer Science
Clustering in interfering binary mixtures
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Delay performance in random-access grid networks
Performance Evaluation
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Many local Markov chains based on Glauber dynamics are known to undergo a phase transition as a parameter λ of the system is varied. For independent sets on the 2-dimensional Cartesian lattice, the Gibbs distribution assigns each independent set a weight λ[I], and the Markov chain adds or deletes a single vertex at each step, It is believed that there is a critical point λc ≈ 3.79 such that for λ c, local dynamics converge in polynomial time while for λ λc they require exponential time. We introduce a new method for showing slow mixing based on the presence or absence of certain topological obstructions in the independent sets. Using elementary arguments, we show that Glauber dynamics will be slow for sampling independent sets in 2 dimensions when λ ≥ 8.066, improving on the best known bound by a factor of 10. We also show they are slow on the torus when λ ≥ 6.183.