Continuity and anomalous fluctuations in random walks in dynamic random environments: numerics, phase diagrams and conjectures

TL;DR

This paper investigates how one-dimensional random walks behave in dynamic environments with different correlation decay rates, revealing new phases and anomalous fluctuations that extend static environment results.

Contribution

It provides numerical analysis and phase diagrams for random walks in dynamic environments, highlighting the occurrence of non-diffusive regimes previously known only in static cases.

Findings

Non-diffusive regimes occur in dynamic environments.

Different behaviors depend on environment dynamics and jump rate ratios.

A new phase diagram for anomalous fluctuations is proposed.

Abstract

We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamic setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments.