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. [K. Ramanan, A. Sengupta, I. Ziedins and P. Mitra, Adv. Appl. Probab., 34 (2002), pp. 1–27] proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the multistate 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)σ(v). In this work, we consider this generalization when G is an infinite rooted b-ary tree and prove rigorously som...