Collision probability for random trajectories in two dimensions

AbstractWe give a lower bound for the non-collision probability up to a long time T in a system of n independent random walks with fixed obstacles on Z2. By ‘collision’ we mean collision between the random walks as well as collision with the fixed obstacles. We give an analogous result for Brownian particles on the plane. As a corollary we show that the non-collision request leads only to logarithmic corrections for a spread-out property of the independent random walk system