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Random Walks in Cooling Random Environments (RWCRE) is a model of randomwalks in dynamic random environments where the environment is frozen between afixed sequence of times (called the cooling map) where it is resampled.Naturally the limiting distributions for this model depend both on thestructure of the cooling sequence and on distribution $\mu$ from which theenvironments are sampled. Previous results have considered the cases where$\mu$ is such that the corresponding model of random walks in a fixed randomenvironment (RWRE) is either (1) recurrent, (2) has a Gaussian limit withdiffusive scaling (the $\kappa > 2$ case), or (3) has positive speed and astable, non-Gaussian limit (the $\kappa \in (1,2)$ case).  In this paper we examine the limiting distributions in two other transientregimes: the sub-ballistic, non-stable regime (i.e., $\kappa \in (0,1)$), andthe Gaussian regime with non-diffusive scaling (i.e., $\kappa = 2$). In thefirst case we show that the limiting distributions are either Gaussian or amixture of Gaussian and independent sums of Mittag-Leffler random variables,while in the second case the limiting distributions are always Gaussian butwith a scaling that differs from the standard deviation by factor (which canoscillate, but which remains confined to some interval $[\beta,1]$) thatdepends very delicately on the properties of the cooling map.

Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime