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

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 arXiv:1906.03167. 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 renormalisation 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.