  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.
