Critical behaviour of the random-field Ising model with long-range interactions in one dimension

We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (alpha r(-(d+sigma))) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of sigma, separated by a critical value sigma(c). What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.