Gaps between avalanches in one-dimensional random-field Ising models

We analyze the statistics of gaps (Delta H) between successive avalanches in one-dimensional random-field Ising models (RFIMs) in an external field H at zero temperature. In the first part of the paper we study the nearest-neighbor ferromagnetic RFIM. We map the sequence of avalanches in this system to a nonhomogeneous Poisson process with an H-dependent rate rho(H). We use this to analytically compute the distribution of gaps P(Delta H) between avalanches as the field is increased monotonically from -infinity to +infinity. We show that P(Delta H) tends to a constant C(R) as Delta H -> 0(+), which displays a nontrivial behavior with the strength of disorder R. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behavior P(Delta H) = 0 up to a system size dependent offset value Delta H-off, and P(Delta H) similar to (Delta H - Delta H-off)(theta) as Delta H -> H-off(+). We perform numerical simulations on this model and determine theta approximate to 0.95(5). We also discuss mechanisms which would lead to a nonzero exponent theta for general spin models with quenched random fields.