The multi-state hard core model on a regular tree

TL;DR

This paper rigorously analyzes a multi-state hard core model on infinite trees, revealing phase transitions and coexistence phenomena depending on the capacity and activity parameters, extending classical models from statistical physics.

Contribution

It proves conjectures about phase transitions in the multi-state hard core model on regular trees, including the nature and location of these transitions for various capacities.

Findings

The C=2 model exhibits a first-order phase transition at higher activity values than the C=1 model.

For large branching factor b, there is an interval of activity values with phase coexistence.

Transition points depend on C being odd or even, with explicit formulas for their locations.

Abstract

The classical hard core model from statistical physics, with activity $\lambda > 0$ and capacity $C=1$, on a graph $G$, concerns a probability measure on the set ${\mathcal I}(G)$ of independent sets of $G$, with the measure of each independent set $I \in {\mathcal I}(G)$ being proportional to $\lambda^{|I|}$. Ramanan et al. proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the {\em 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 $\{0,\ldots,C\}$ 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 $\lambda^{i}$, so that the probability of a configuration $\sigma:V(G)\rightarrow \{0,\ldots, C\}$ is proportional to $\lambda^{\sum_{v \in V(G)} \sigma(v)}$. 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 $\lambda$ than the $C=1$ model exhibits its (second-order) phase transition. In addition, for large $b$ we identify a short interval of values for $\lambda$ 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 $\lambda = (e/b)^{1/\ceil{C/2}}$, while for even $C$, it occurs around $\lambda=(\log b/b(C+2))^{2/(C+2)}$. In the latter case, the transition is first-order.