Aging in Metropolis dynamics of the REM: a proof

TL;DR

This paper rigorously proves aging phenomena in the REM under Metropolis dynamics, showing convergence of correlation functions to the arcsine law and establishing almost sure convergence of a rescaled clock process to a stable subordinator, refining previous results.

Contribution

It provides a rigorous proof of aging in the REM with Metropolis dynamics, including almost sure convergence results and refinement of earlier conjectures.

Findings

Correlation functions converge to the arcsine law

Rescaled clock process converges almost surely to a stable subordinator

Refinement and proof of a conjecture from prior work

Abstract

We study the aging behavior of the Random Energy Model (REM) evolving under Metropolis dynamics. We prove that a classical two-time correlation function converges almost surely to the arcsine law distribution function that characterizes activated aging, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. In the course of the proof we establish that a certain continuous time clock process, after proper rescaling, converges almost surely to a stable subordinator, improving upon the result of Cerny and Wassmer (2015) where a closely related clock is shown to converge in probability only, and in a restricted region of the time-scale and temperature parameters. The random rescaling involved in this convergence is controlled at the fine level of fluctuations. As a byproduct, we refine and prove a conjecture made in Cerny and Wassmer (2015).