Random walks in cooling random environments

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is resampled at every unit of time; (II) the random environment is sampled at time zero only (i.e. the well-known RWRE model where the environment is static/frozen). In our model the random environment is resampled along a determinstic sequence of refreshing times.

Depending on the choice of these refreshing times the resulting long-term behavior can be close or not to the two interpolated models. We will make this clear as far as recurrence, asymptotic speed, fluctuations and large deviations are concerned. For those questions we show the emergence of a richer palette of behaviors with respect to the two interpolated models. In particular we will identify scenarios where either homogenization or localization occur, giving rise to qualitative different asymptotics, e.g. classical diffusive Gaussian regimes versus sub-diffusive regimes with mixed limiting laws.

The general philosophy is to explore how a well-understood stochastic process with a rich correlation structure gets affected by adding noise through local independence.

Joint work with Y. Chino, C. da Costa and F. den Hollander