Hitting times for independent random walks on $\mathbb{Z}^d$

Abstract

We consider a system of asymmetric independent random walks on $\mathbb{Z}^d$, denoted by $\{η_t,t\in{\mathbb{R}}\}$, stationary under the product Poisson measure $ν_ρ$ of marginal density $ρ>0$. We fix a pattern $\mathcal{A}$, an increasing local event, and denote by $τ$ the hitting time of $\mathcal{A}$. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of $η_t$ conditioned on $\{τ>t\}$ for all times $t$. When $d\ge3$, this provides bounds on the rate of convergence of the law of $η_t$ conditioned on $\{τ>t\}$ toward its limiting probability measure as $t$ tends to infinity. We also treat the case where the initial measure is close to $ν_ρ$ without being product.