Cover times for Brownian motion and random walks in two dimensions

Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni
2004 Annals of Mathematics  
Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x∈T 2 T (x, ε)/| log ε| 2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, twodimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to , that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 n is asymptotic
more » ... Z 2 n is asymptotic to 4n 2 (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 non-rigorously 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 Z 2 to cover the disc of radius n. Date
doi:10.4007/annals.2004.160.433 fatcat:jh5ds6z2lbbchlbzlbluh6uexy