Asymptotic behavior and aging of a low temperature cascading 2-GREM dynamics at extreme time scales

TL;DR

This paper investigates the long-term dynamics and aging phenomena of a low-temperature cascading 2-GREM model, revealing multiple dynamical phases and their scaling limits at extreme time scales.

Contribution

It introduces new scaling limit results and aging behaviors for the cascading 2-GREM at low temperatures, especially at fine tuning temperatures near critical points.

Findings

Three distinct dynamical phases below the lowest critical temperature.

Limiting dynamics characterized by K processes depending on temperature regimes.

Aging phenomena observed in the limiting processes at extreme time scales.

Abstract

We derive scaling limit results for the 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.