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 $d\ge3$, 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.