Collision probability for random trajectories in two dimensions

We 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 . 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.