Hitting times for independent random walks on ℤd

Abstract

We consider a system of asymmetric independent random walks on Z d , denoted by {η t . t e R}, stationary under the product Poisson measure V p of marginal density p > 0. We fix a pattern A, an increasing local event, and denote by τ the hitting time of 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 (r > t} for all times t. When d ≥ 3, this provides bounds on the rate of convergence of the law of η t conditioned on (r > t} toward its limiting probability measure as t tends to infinity. We also treat the case where the initial measure is close to V ρ without being product.