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October 2016 Second-order Markov random fields for independent sets on the infinite Cayley tree
David A. Goldberg
Ann. Appl. Probab. 26(5): 2626-2660 (October 2016). DOI: 10.1214/15-AAP1156

Abstract

Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e., hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise, for example, in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interactions on the independent sets of the infinite Cayley tree. We prove that the associated Gibbsian specification satisfies the celebrated FKG inequality whenever the local potentials defining the Hamiltonian satisfy a log-convexity condition. Under this condition, we give necessary and sufficient conditions for the existence of a unique infinite-volume Gibbs measure in terms of an explicit system of equations, prove the existence of a phase transition and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For potentials which are small perturbations of those coinciding to the hardcore model at the critical activity, we characterize whether the resulting specification has a unique infinite-volume Gibbs measure in terms of whether these perturbations satisfy an explicit linear inequality. Our analysis reveals an interesting nonmonotonicity with regards to biasing toward excluded nodes with no included neighbors.

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David A. Goldberg. "Second-order Markov random fields for independent sets on the infinite Cayley tree." Ann. Appl. Probab. 26 (5) 2626 - 2660, October 2016. https://doi.org/10.1214/15-AAP1156

Information

Received: 1 October 2013; Revised: 1 June 2015; Published: October 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1353.60087
MathSciNet: MR3563189
Digital Object Identifier: 10.1214/15-AAP1156

Subjects:
Primary: 60K35

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 5 • October 2016
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