Hysteresis in the random-field Ising model and bootstrap percolation

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging Lstar increases as exp[exp(J/Delta)] in 2D, and as exp{exp[exp(J/Delta )]} in 3D, for disorder strength Delta much less than the exchange coupling J . For system size 1infinity for both square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and h(coer) tends to J .