Mathematics – Probability
Scientific paper
2012-02-20
Mathematics
Probability
46 pages, 3 figures. Companion paper to arXiv:0909.4320
Scientific paper
The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the $d$-dimensional torus $(Z/nZ)^d$ for any $d\geq 1$. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for stochastic Ising on any sequence of bounded-degree graphs with sub-exponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for non-monotone spin-systems at high temperatures, including the gas hard-core model, the Potts model, the anti-ferromagnetic Potts model and the coloring model.
Lubetzky Eyal
Sly Allan
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