Abstract
We derive scaling limit results for the Hierarchical Random Hopping Dynamics for the cascading two-level GREM at low temperature at extreme time scales. It is known that in the cascading regime there are two static critical temperatures. We show that there exists a (narrow) set of fine tuning temperatures; when they lie below the static lowest critical temperature, three distinct dynamical phases emerge below the lowest critical temperature, with three different types of limiting dynamics depending on whether the temperature is (well) above or below, or at a fine tuning temperature, all of which are given in terms of K processes. We also derive scaling limit results for temperatures between the lowest and he highest critical ones, as well as aging results for all the limiting processes mentioned above, by taking a second small time limit.
Highlights
It is believed that activated dynamics of spin glasses, which occur at time-scales on which the process has time to escape deep valleys of a highly complex random energy landscape, exhibit aging at low temperature, the nature of which should be closely linked to the specific properties of this landscape [10], [9]
Would a physically more realistic choice of dynamics change the nature of aging? The recent paper [24] answered in the negative for the key example of Metropolis dynamics of the REM, for all time-scales away from extreme time-scales at which the phase transition from aging to stationarity takes place
Let us briefly anticipate our main aging results, holding in the case described at the last paragraph of Subsection 1.2.1, where fine tuning temperatures are below the lowest static critical temperature
Summary
It is believed that activated dynamics of spin glasses, which occur at time-scales on which the process has time to escape deep valleys of a highly complex random energy landscape, exhibit aging at low temperature, the nature of which should be closely linked to the specific properties of this landscape [10], [9]. Do we obtain a new, distinctive GREM-like aging behavior that goes beyond the known REM-like behavior, but we isolate a new, temperature dependent fine tuning mechanism, that gives rise to three distinct aging regimes (corresponding to three distinct K-processes) which the dynamics can be tuned in by adjusting the temperature. This completely new fine tuning mechanism, and the rich aging picture that emerges were not predicted in the physics literature on the GREM dynamics [10], [29], [9]
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