
TL;DR
This paper investigates the probability and distribution of local configurations in high-dimensional binary Markov random fields on large lattices, providing bounds and conditions for their occurrence.
Contribution
It introduces precise probabilistic bounds and conditions for local configuration occurrences in large-scale Markov random fields with size-dependent potentials.
Findings
Bounds for local configuration probabilities in large lattices
Estimates of distances between configurations based on weights
Conditions for universal occurrence of configurations
Abstract
A -dimensional binary Markov random field on a lattice torus is considered. As the size of the lattice tends to infinity, potentials and depend on . Precise bounds for the probability for local configurations to occur in a large ball are given. Under some conditions bearing on and , 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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