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.


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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