Critical exponents of the random field hierarchical model

We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, J(r)∼r-ρ, and we can explore mean-field and non-mean-field behavior by changing ρ. We analyze the model at T=0 and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent δ, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in D-dimensional short range models and with the ones of the straightforward one-dimensional long range model.