Convergence of clock processes and aging in Metropolis dynamics of a truncated REM

TL;DR

This paper demonstrates that Metropolis dynamics of a truncated REM exhibit aging behavior similar to Bouchaud's trap model, with convergence to stable subordinators and an arcsine law for correlation functions.

Contribution

It introduces a novel clock process convergence approach where the number of summands is itself a clock, extending universality results for aging in spin glasses.

Findings

Correlation functions converge to an arcsine law.

Clock processes converge to stable subordinators.

Aging behavior is universal across models.

Abstract

We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud's REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence result of a new type where the number of summands is itself a clock process. This reflects the fact that the exploration process of Metropolis dynamics is itself an aging process, governed by its own clock. Both clock processes are shown to converge to stable subordinators below certain critical lines in their time-scale and temperature domains, almost surely in the random environment.