Replica-symmetry-breaking transitions and off-equilibrium dynamics

I consider branches of replica-symmetry-breaking (RSB) solutions in glassy systems that display a dynamical transition at a temperature T-d characterized by a mode-coupling-theory dynamical behavior. Below T-d these branches of solutions are considered to be relevant to the system complexity and to off-equilibrium dynamics. Under general assumptions I argue that near T-d it is not possible to stabilize the one-step (1RSB) solution beyond the marginal point by making a full RSB (FRSB) ansatz. However, depending on the model, there may exist a temperature T-* strictly lower than T-d below which the 1RSB branch can be continued to a FRSB branch. Such a temperature certainly exists for models that display the so-called Gardner transition and in this case T-G < T-* < T-d. An analytical study in the context of the truncated model reveals that the FRSB branch of solutions below T-* is characterized by a two-plateau structure and it ends where the first plateau disappears. These general features are confirmed in the context of the Ising p-spin model with p = 3 by means of a numerical solution of the FRSB equations. The results are discussed in connection with off-equilibrium dynamics within Cugliandolo-Kurchan theory. In this context I assume that the RSB solution relevant for off-equilibrium dynamics is the 1RSB marginal solution in the whole range (T-*, T-d) and it is the end point of the FRSB branch for T < T-*. Remarkably, under these assumptions it can be argued that T-* marks a qualitative change in off-equilibrium dynamics in the sense that the decay of various dynamical quantities changes from power law to logarithmic.