Extremal Aging For Trap Models

TL;DR

This paper introduces a new extremal aging mechanism for trap models, extending the classical stable subordinator framework to include extremal processes, and applies it to various spin glass models and random walks.

Contribution

It proposes extremal processes as a novel aging mechanism, extending the existing stable subordinator approach to a broader class of models and scales.

Findings

Extremal aging mechanism applies to REM, SK, p-spin models, and biased random walks.

The mechanism explains aging behavior at various temperature and time scales.

It extends the universality class of aging phenomena beyond stable subordinators.

Abstract

In the seminal work [5], Ben Arous and \v{C}ern\'y give a general characterization of aging for trap models in terms of $\alpha$-stable subordinators with $\alpha \in (0,1)$. Some of the important examples that fall into this universality class are Random Hopping Time (RHT) dynamics of Random Energy Model (REM) and $p$-spin models observed on exponential time scales. In this paper, we explain a different aging mechanism in terms of {\it extremal processes} that can be seen as the extension of $\alpha$-stable aging to the case $\alpha=0$. We apply this mechanism to the RHT dynamics of the REM for a wide range of temperature and time scales. The other examples that exhibit extremal aging include the Sherrington Kirkpatrick (SK) model and $p$-spin models [6, 9], and biased random walk on critical Galton-Watson trees conditioned to survive [11].