Cover times for Brownian motion and random walks in two dimensions

Let T(x, epsilon) denote the first hitting time of the disc of radius epsilon centered at x for Brownian motion on the two dimensional torus T-2. We prove that SUPx epsilon T2 T(x, epsilon)/vertical bar log epsilon vertical bar(2) - 2/pi as epsilon -> 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z(n)(2) is asymptotic to 4n(2)(logn)(2)/pi. Den termining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also establish a conjecture, due to Kesten and Revesz, that describes the asymptotics for the number of steps needed by simple random walk in Z(2) to cover the disc of radius n.