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Mixing time bounds for oriented kinetically constrained spin models. (English) Zbl 1329.60332
Summary: We analyze the mixing time of a class of oriented kinetically constrained spin models (KCMs) on a $$d$$-dimensional lattice of $$n$$ sites. A typical example is the North-East model, a $$0-1$$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site’s southerly and westerly nearest neighbours have spin 0, with rate one it resets its own spin by tossing a $$p$$-coin, at all other times its spin remains frozen. Such models are very popular in statistical physics because, in spite of their simplicity, they display some of the key features of the dynamics of real glasses. We prove that the mixing time is $$O(n \log n)$$ whenever the relaxation time is $$O(1)$$. Our study was motivated by the “shape” conjecture put forward by G. Kordzakhia and S. P. Lalley [J. Appl. Probab. 43, No. 3, 782–792 (2006; Zbl 1134.60060)].

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J27 Continuous-time Markov processes on discrete state spaces 60J28 Applications of continuous-time Markov processes on discrete state spaces
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