  In previous work by Avena and den Hollander, a model of a one-dimensional
random walk in a dynamic random environment was proposed where the random
environment is resampled from a given law along a growing sequence of
deterministic times. In the regime where the increments of the resampling times
diverge, which is referred to as the cooling regime, a weak law of large
numbers and certain fluctuation properties were derived under the annealed
measure. In the present paper we show that a strong law of large numbers and a
quenched large deviation principle hold as well. In the cooling regime, the
random walk can be represented as a sum of independent variables, distributed
as the increments of a random walk in a static random environment over
increasing periods of time. Our proofs require suitable multi-layer
decompositions of sums of random variables controlled by moments bounds and
concentration estimates. Along the way we derive two results of independent
interest, namely, a concentration inequality for the cumulants of the
displacement in the static random environment and an ergodic theorem that deals
with limits of sums of triangular arrays representing the structure of the
cooling regime. We close by discussing our present understanding of
homogenisation effects as a function of the speed of divergence of the
increments of the resampling times.
