Random-field Ising model on complete graphs and trees

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We present exact results for the critical behavior of the RFIM on complete graphs and trees, both at equilibrium and away from equilibrium, i.e., models for hysteresis and Barkhausen noise. We show that for stretched exponential and power law distributions of random fields the behavior on complete graphs is non-universal, while the behavior on Cayley trees is universal even in the limit of large co-ordination.