Collective degrees of freedom and multiscale dynamics in spin glasses

We relate the long-range, long-time correlations during the simulation of disordered complex systems to the relevant macroscopic effective collective degrees of freedom. We prove that in systems that have an ultrametric space of ground states, the tunneling between vacuums cannot be expressed in terms of spatially disjoint clusters or in terms of spatial multiscale hierarchies. We relate this to the ultraslow convergence difficulties of multiscale-cluster algorithms in such systems. On the contrary, in the case of finite connectivity (dilute) spin glasses, we are able to find multiscale-cluster algorithms that are much more efficient than the usual methods. We relate their efficiency explicitly to their action on specific collective degrees of freedom. These degrees of freedom are responsible for the slowing down of the usual algorithms.