Sharp threshold for the ballisticity of the random walk on the exclusion process

Abstract

We study a non-reversible random walk advected by the symmetric simple exclusion process, so that the walk has a local drift of opposite sign when sitting atop an occupied or an empty site. We prove that the back-tracking probability of the walk exhibits a sharp transition as the density \rho of particles in the underlying exclusion process varies across a critical density \rho_{c} . Our results imply that the speed v=v(\rho) of the walk is a strictly monotone function and that the zero-speed regime is either absent or collapses to a single point, \rho_{c} , thus solving a conjecture of Hilário et al. (2020). The proof proceeds by exhibiting a quantitative monotonicity result for the speed of a truncated model, in which the environment is renewed after a finite time horizon L . The truncation parameter L is subsequently pitted against the density \rho to carry estimates over to the full model. Our strategy is somewhat reminiscent of certain techniques recently used to prove sharpness results in percolation problems. A key instrument is a combination of renormalization arguments with refined couplings of environments at slightly different densities, which we develop in this article. Our results hold in fact in greater generality and apply to a class of environments with possibly egregious features, outside perturbative regimes.