The Multistate Hard Core Model on a Regular Tree

The classical hard core model from statistical physics, with activity and capacity , on a graph , concerns a probability measure on the set of independent sets of , with the measure of each independent set being proportional to . 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 is allowed to be a positive integer, and a configuration in the model is an assignment of states from to (the set of nodes of ) subject to the constraint that the states of adjacent nodes may not sum to more than . The activity associated to state is , so that the probability of a configuration is proportional to . In this work, we consider this generalization when is an infinite rooted -ary tree and prove rigorously some of the conjectures made by Ramanan et al. In particular, we show that the model exhibits a (first-order) phase transition at a larger value of than the model exhibits its (second-order) phase transition. In addition, for large we identify a short interval of values for above which the model exhibits phase coexistence and below which there is phase uniqueness. For odd , this transition occurs in the region of , while for even , it occurs around . In the latter case, the transition is first-order.

Перевод пока недоступен