## Hitting times for independent random walks on $$\mathbb Z^d$$.(English)Zbl 1101.60074

Summary: 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 $$d\geq 3$$, 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.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 60J25 Continuous-time Markov processes on general state spaces
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