Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, due to t he effect of singularities in the interactions between fields involving sev eral replicas below dimension eight. In the zero-replica limit, several cou pling constants have thus to be considered, instead of just one. As a resul t we found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, whi ch does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Si milarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin to a t which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complet e correspondence between the equilibrium singularities in the limit at n = 0, and the singularities in dynamics when the initial time to goes to minus infinity, with n replaced by -1/t(0). There is thus complete coherence bet ween the two approaches. (C)Academie des sciences/Elsevier, Paris.