  Random Walks in Cooling Random Environments (RWCRE) is a model of random
walks in dynamic random environments where the entire environment is resampled
along a fixed sequence of times, called the "cooling sequence," and is kept
fixed in between those times. This model interpolates between that of a
homogenous random walk, where the environment is reset at every step, and
Random Walks in (static) Random Environments (RWRE), where the environment is
never resampled. In this work we focus on the limiting distributions of
one-dimensional RWCRE in the regime where the fluctuations of the corresponding
(static) RWRE is given by a $s$-stable random variable with $s\in(1,2)$. In
this regime, due to the two extreme cases (resampling every step and never
resampling, respectively), a crossover from Gaussian to stable limits for
sufficiently regular cooling sequence was previously conjectured. Our first
result answers affirmatively this conjecture by making clear critical exponent,
norming sequences and limiting laws associated with the crossover which
demonstrates a change from Gaussian to $s$-stable limits, passing at
criticality through a certain generalized tempered stable distribution. We then
explore the resulting RWCRE scaling limits for general cooling sequences. On
the one hand, we offer sets of operative sufficient conditions that guarantee
asymptotic emergence of either Gaussian, $s$-stable or generalized tempered
distributions from a certain class. On the other hand, we give explicit
examples and describe how to construct irregular cooling sequences for which
the corresponding limit law is characterized by mixtures of the three above
mentioned laws. To obtain these results, we need and derive a number of refined
asymptotic results for the static RWRE with $s\in(1,2)$ which may be of
independent interest.
