Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices

Details Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices

2005-08-25

arxiv.org/abs/cond-mat/0508609v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

10 pages, 3 figures

Scientific paper

10.1103/PhysRevE.72.066127

We study XY spin systems on small world lattices for a variety of graph structures, e.g. Poisson and scale-free, superimposed upon a one dimensional chain. In order to solve this model we extend the cavity method in the one pure-state approximation to deal with real-valued dynamical variables. We find that small-world architectures significantly enlarge the region in parameter space where synchronization occurs. We contrast the results of population dynamics performed on a truncated set of cavity fields with Monte Carlo simulations and find excellent agreement. Further, we investigate the appearance of replica symmetry breaking in the spin-glass phase by numerically analyzing the proliferation of pure states in the message passing equations.

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