Geography of local configurations

Details Geography of local configurations Geography of local configurations

2007-07-19

arxiv.org/abs/0707.2889v2

Annals of Applied Probability 2010, Vol. 20, No. 3, 806-840

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP630 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP630

A $d$-dimensional binary Markov random field on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, potentials $a=a(n)$ and $b=b(n)$ depend on $n$. Precise bounds for the probability for local configurations to occur in a large ball are given. Under some conditions bearing on $a(n)$ and $b(n)$, the distance between copies of different local configurations is estimated according to their weights. Finally, a sufficient condition ensuring that a given local configuration occurs everywhere in the lattice is suggested.

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