  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 e.g. 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 non-monotonicity with
regards to biasing towards excluded nodes with no included neighbors.
