Sarah De Nigris, Xavier Leoncini
We study the XY -rotors model on small networks whose number of links scales with the system size N_{links}\sim N^{\gamma}, where 1\le\gamma\le2 . We first focus on regular one dimensional rings. For \gamma<1.5 the model behaves like short-range one and no phase transition occurs. For \gamma>1.5, the system equilibrium properties are found to be identical to the mean field, which displays a second order phase transition at \epsilon_{c}=0.75 . Moreover for \gamma_{c}=1.5 we find that a non trivial state emerges, characterized by an infinite susceptibility. We then consider small world networks, using the Watts-Strogatz mechanism on the regular networks parametrized by \gamma . We first analyze the topology and find that the small world regime appears for rewiring probabilities which scale as p_{SW}\propto1/N^{\gamma} . Then considering the XY -rotors model on these networks, we find that a second order phase transition occurs at a critical energy \varepsilon_{c} which logarithmically depends on the topological parameters p and \gamma . We also define a critical probability p_{MF}, corresponding to the probability beyond which the mean field is quantitatively recovered, and we analyze its dependence on \gamma .
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http://arxiv.org/abs/1304.4854
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