Geography of local configurations

David Coupier

doi:10.1214/09-AAP630

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Home > Journals > Ann. Appl. Probab. > Volume 20 > Issue 3 > Article

June 2010 Geography of local configurations

David Coupier

Ann. Appl. Probab. 20(3): 806-840 (June 2010). DOI: 10.1214/09-AAP630

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Abstract

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.