Aging in Metropolis dynamics of the REM: a proof

Understand

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.