A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics

Details A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics

2011-07-15

arxiv.org/abs/1107.2970v1

Mathematics

Probability

87 pages

Scientific paper

The Swendsen-Wang dynamics is a Markov chain widely used by physicists to sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer, Frieze and Rue proved that on the complete graph K_n the mixing time of the chain is at most O(n^{1/2}) for all non-critical temperatures. In this paper we show that the mixing time is Theta(1) in high temperatures, Theta(log n) in low temperatures and Theta(n^{1/4}) at criticality. We also provide an upper bound of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on any tree with n vertices.

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