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The multi-state hard core model on a regular tree | | David Galvin
; Fabio Martinelli
; Kavita Ramanan
; Prasad Tetali
; | | Date: |
27 Jul 2010 | | 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 $sigma:V(G)
ightarrow {0,ldots, C}$ 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. | | Source: | arXiv, 1007.4806 | | Services: | Forum | Review | PDF | Favorites | |
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