The Multistate 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|. Ramanan et al. proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the multi-state hard core model, the capacity C is allowed to be a positive integer, and a configuration in the model is an assignment of states from {0, . . . , C} to V (G) (the set of nodes of G) subject to the constraint that the states of adjacent nodes may not sum to more than C. The activity associated to state i is λi, so that the probability of a configuration σ : V (G) → {0, . . . , C} is proportional to λ ∑ v∈V (G) . In this work, we consider this generalization when G is an infinite rooted b-ary tree and prove rigorously some of the conjectures made by Ramanan et al. In particular, we show that the C = 2 model exhibits a (first-order) phase transition at a larger value of λ than the C = 1 model exhibits its (second-order) phase transition. 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/⌈C/2⌉, while for even C, it occurs around λ = (log b/b(C + 2))2/(C+2). In the latter case, the transition is first-order.