Random Walk on a Perturbation of the Infinitely-Fast Mixing Interchange Process

We consider a random walk in dimension in a dynamic random environment evolving as an interchange process with rate . We prove that, if we choose large enough, almost surely the empirical velocity of the walker eventually lies in an arbitrary small ball around the annealed drift. This statement is thus a perturbation of the case where the environment is refreshed between each step of the walker. We extend three-way part of the results of Huveneers and Simenhaus (Electron J Probab 20(105):42, 2015), where the environment was given by the 1-dimensional exclusion process: (i) We deal with any dimension ; (ii) We treat the much more general interchange process, where each particle carries a transition vector chosen according to an arbitrary law ; (iii) We show that is not only in the same direction of the annealed drift, but that it is also close to it.