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

We consider a system of asymmetric independent random walks on ${\mathbb{Z}}^d$, denoted by $\{{\eta }_t,t\in \mathbb{R}\}$, stationary under the product Poisson measure ${\nu }_{\rho }$ of marginal density $\rho >0$. We fix a pattern $\mathcal{A}$, an increasing local event, and denote by $\tau$ 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 ${\eta }_t$ conditioned on $\{\tau >t\}$ for all times $t$. When $d\ge 3$, this provides bounds on the rate of convergence of the law of ${\eta }_t$ conditioned on $\{\tau >t\}$ toward its limiting probability measure as $t$ tends to infinity. We also treat the case where the initial measure is close to ${\nu }_{\rho }$ without being product.