The random walk on dynamic environments, such as the simple exclusion process (SEP) on a d-dimensional lattice, has attracted considerable attention in the last decade. In this model, a random walker moves at each unit of time to a neighbouring vertex, with a drift that depends on whether its current location is occupied or not by a SEP particle.

We will focus on the case d=1, when occupied sites have a drift to the right, and empty sites a drift to the left. The parameter of interest will be the density of particles: when it is large (resp. small), the random walk has a positive (resp. negative) speed.

Since the SEP is conservative and mixes slowly, a natural question is whether there are strong trapping effects, as in the more classical setting of static environments. Could it be for instance that for a non-empty interval of intermediate densities, the random walk has zero speed? We give a negative answer, showing that the speed is strictly increasing with the density.

The proof uses a comparison with a finite-range model (via renormalisation), and an original coupling to circumvent the bad mixing properties of the SEP.

This is joint work with Daniel Kious and Pierre-François Rodriguez.

**Date**: 20 May 2024, 14:00 (Monday, 5th week, Trinity 2024)**Venue**: Venue to be announced**Speaker**: Guillaume Conchon-Kerjan (King College London)**Organising department**: Department of Statistics**Organisers**: Matthias Winkel (Department of Statistics, University of Oxford), Julien Berestycki (University of Oxford), Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Members of the University only- Editor: Julien Berestycki