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

Nikos S Skantzos·Isaac Pérez Castillo·Jonathan P L Hatchett
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Abstract

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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Nikos S Skantzos, Isaac Pérez Castillo, Jonathan P L Hatchett. 2005-08-25. Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices. https://doi.org/10.1103/physreve.72.066127

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