The Finite-state Hard Core Model on a Regular Tree

The classical hard core model from statistical physics, with activity λ > 0 and capacity C = 1, on a graph G, concerns a probability measure on the set I(G) of independent sets of G, with the measure of each independent set I ∈ I(G) being proportional to λ|I|. When G is a (b + 1)-regular tree, a generalization of the hard core model that has capacity C larger than one was studied by Mitra et al., as an idealized model of multicasting in communication networks. In this work, we consider this generalization and prove rigorously some of the conjectures made by Mitra et al. In particular, we show that the nature of the phase transition indeed depends on the parity of the capacity parameter C ∈ Z+. In addition, for large b we identify a short interval of values for λ above which the model exhibits phase co-existence and below which there is phase uniqueness. For odd C, this transition occurs in the region of λ = (e/b)1/dC/2e, while for even C, it occurs around λ = (log b/b(C + 2)).