The multi-state hard core model on a regular tree

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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₎ σ⁽ᵛ⁾. 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=(b/b(C+2))²/⁽ᶜ⁺²⁾. In the latter case, the transition is first-order.