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Recently, there has been significant interest in understanding the propertiesof Markov random fields (M.r.f.) defined on the independent sets of sparsegraphs. When these M.r.f. are restricted to pairwise interactions (i.e.hardcore model), much progress has been made. However, considerably less isknown in the presence of higher-order interactions, which arise e.g. in theanalysis of independent sets with special properties and the study ofresource-constrained communication networks. In this paper, we further ourunderstanding of such models by analyzing M.r.f. with second-order interactionson the independent sets of the infinite Cayley tree. We prove that theassociated Gibbsian specification satisfies the celebrated FKG Inequalitywhenever the local potentials defining the Hamiltonian satisfy a log-convexitycondition. Under this condition, we give necessary and sufficient conditionsfor the existence of a unique infinite-volume Gibbs measure in terms of anexplicit system of equations, prove the existence of a phase transition, andgive explicit bounds on the associated critical activity, which we prove toexhibit a certain robustness. For potentials which are small perturbations ofthose coinciding to the hardcore model at the critical activity, wecharacterize whether the resulting specification has a unique infinite-volumeGibbs measure in terms of whether these perturbations satisfy an explicitlinear inequality. Our analysis reveals an interesting non-monotonicity withregards to biasing towards excluded nodes with no included neighbors.

Second-order Markov random fields for independent sets on the infinite Cayley tree