Geography of local configurations

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 $d$-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.