Cover times for Brownian motion and random walks in two dimensions

Abstract

Let τ(x, ε) denote the first hitting time of the disc of radius e centered at x for Brownian motion on the two dimensional torus double-struck T sign2. We prove that supx∈T2 τ(x, ε)/|log epsi;|2 → 2/π as → 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 ℤn2 is asymptotic to 4n2(log n)2/π. Determining 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 Révész, that describes the asymptotics for the number of steps needed by simple random walk in ℤ2 to cover the disc of radius n.