Mixing time for the Ising model: a uniform lower bound for all graphs

Details Mixing time for the Ising model: a uniform lower bound for all graphs Mixing time for the Ising model: a uniform lower bound for all graphs

2009-09-28

arxiv.org/abs/0909.5162v1

Mathematics

Probability

11 pages

Scientific paper

Consider Glauber dynamics for the Ising model on a graph of $n$ vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least $n\log n/f(\Delta)$, where $\Delta$ is the maximum degree and $f(\Delta) = \Theta(\Delta \log^2 \Delta)$. Their result applies to more general spin systems, and in that generality, they showed that some dependence on $\Delta$ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any $n$-vertex graph is at least $(1/4+o(1))n \log n$.

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