Collision probability for random trajectories in two dimensions

We give a lower bound for the non-collision probability up to a long time TT in a system of nn independent random walks with fixed obstacles on Z2Z2. 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.