We study a random walk on the $d$-dimensional lattice ($d\geq 5$) whose (discrete-time) movements are randomly driven by an underlying symmetric Simple Exclusion Process (SEP). In this set-up, standard techniques from static environments no longer apply. Moreover, the SEP is conservative and induces long-range space-time correlations. With the loss of monotonicity in dimensions above one, only perturbative results have been derived thus far.
We prove a law of large numbers in a wide regime of parameters. To do so, we track the measure of the environment conditioned on the past trajectory of the walker. This measure can be bounded in some strong sense between two inhomogeneous products of Bernoulli measures, that differ from the homogeneous one in a ‘summable’ way if the dimension is large enough. Joint work with Daniel Kious and Rémy Poudevigne.
