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, by the effect of singularities in the interactions between fields involving several replicas, below dimension eight. In the zero-replica limit several coupling constants have thus to be considered, instead of just one. As a result we had 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, which 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. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin $t_0$ at 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 complete correspondence between the equilibrium singularities in the $n=0$ limit, and the singularities in the dynamics when the initial time $t_0$ goes to minus infinity, with $n$ replaced by $-\frac{1}{t_0}$. There is thus complete coherence between the two approaches.