Aging at shorter pre-equilibrium time scales

Determine the aging behavior of the continuous-time random walk in the discrete Gaussian Free Field potential at time scales substantially smaller than the pre-equilibrium scales t_N(\gamma), particularly at scales of order o(N^2), by characterizing the effective potential induced by the DGFF restricted to the range explored by the walk.

Background

The paper establishes aging for time scales t_N(\gamma) with \gamma in (0,1), and explains that these are the longest possible pre-equilibrium scales because the case \gamma=0 is already in equilibrium, where aging does not occur. The authors therefore identify substantially shorter time scales as an unresolved regime.

At scales of order o(N2), the underlying simple random walk does not explore the full torus. Consequently, the relevant random environment is not the DGFF on the entire finite box but the DGFF restricted to the random range of the walk. Understanding this effective potential is presented as the principal obstacle to determining whether and how aging persists, and whether the Generalized Arcsine-law parameter varies across these shorter scales, as it does in the Random Energy Model.

References

An interesting question therefore concerns the behavior of the system at much smaller time scales. In the case of the REM, for example, aging is exhibited over a much larger spectrum of times, with the parameter of the Generalized Arcsine Law varying according to the scale chosen. The difficulty here is that at time scales of order o(N2) the walk does not get the chance to explore the full torus, and thus the effective potential for the problem is the DGFF restricted to the range of the walk. This is a non-trivial object, which is not at all understood.

— Aging in a spin glass with a logarithmic potential  (2608.13361 - Cortines et al., 13 Aug 2026) in Section 1, Subsection “Shorter time scales”