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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 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, it 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 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 1RBS 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.