RANDOM WALK ON A PERTURBATION OF THE INFINITELY-FAST MIXING INTERCHANGE PROCESS

Michele Salvi (1) , François Simenhaus (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider a random walk in dimension d ≥ 1 in a dynamic random environment evolving as an interchange process with rate γ > 0. We only assume that the annealed drift is non–zero. We prove that, if we choose γ large enough, almost surely the empirical velocity of the walker Xt/ t eventually lies in an arbitrary small ball around the annealed drift. This statement is thus a perturbation of the case γ = +∞ where the environment is refreshed between each step of the walker. We extend three-way part of the results of [HS15], where the environment was given by the 1–dimensional exclusion process: (i) We deal with any dimension d ≥ 1; (ii) We treat the much more general interchange process, where each particle carries a transition vector chosen according to an arbitrary law µ; (iii) We show that X t t is not only in the same direction of the annealed drift, but that it is also close to it. AMS subject classification (2010 MSC): 60K37, 82C22, 60Fxx, 82D30.

Mots clés: interchange process; dynamic random environment; renormalisation; limit theorems; Random walk

Domaines: Probabilités [math.PR]

Michele Salvi, François Simenhaus. RANDOM WALK ON A PERTURBATION OF THE INFINITELY-FAST MIXING INTERCHANGE PROCESS. Journal of Statistical Physics, In press. ⟨hal-01619895⟩