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

TL;DR

This paper analyzes the asymptotic behavior of a non-cascading 2-GREM model at low temperatures, revealing three dynamical regimes and how energy landscape exploration varies across time scales.

Contribution

It derives the scaling limit for the Hierarchical Random Hopping dynamics of the non-cascading 2-GREM at low temperatures, highlighting differences from the cascading case.

Findings

Identifies three distinct dynamical regimes based on parameters.

Shows fluctuations of energy pairs scale with volume, affecting dynamics.

Reveals partial exploration of the energy landscape at certain time scales.

Abstract

We derive the scaling limit for the Hierarchical Random Hopping dynamics for the non cascading 2-GREM at low temperatures and time scales where the dynamics is close to equilibrium. The {\em fine tuning} phenomenon plays a role (under certain choices of parameters of the model), yielding three dynamical regimes. In contrast to the {\em cascading} case, the pairs of first and second level energies have fluctuations which scale with the volume, and this leads to a family of time scales where we see the dynamics moving through only a part of the full low temperature energy landscape.