Functional Weak Limit of Random Walks in Cooling Random Environment

TL;DR

This paper establishes the weak limit behavior of random walks in cooling random environments, showing convergence to a time-rescaled Brownian motion under slow cooling and a degenerate limit under fast cooling.

Contribution

It provides the first analysis of the weak limit of RWCRE trajectories under different cooling regimes, extending previous law of large numbers and Gaussian fluctuation results.

Findings

Weak limit is a time-rescaled Brownian motion under slow cooling.

Limit degenerates to a constant in fast cooling.

Results apply to recurrent static environments like Sinai's model.

Abstract

We prove an annealed weak limit of the trajectory of the random walks in cooling random environment (RWCRE) under both slow (polynomial) and fast (exponential) cooling. We identify the weak limit when the underlying static environment is recurrent (Sinai's model). Avena and den Hollander have previously proved law of large numbers and Gaussian fluctuation of RWCRE. We find that the weak limit of the trajectory exists as a time-rescaled Brownian motion in the slow cooling case but the limit degenerates to a constant function in the fast cooling case.