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

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 fair 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.