Cover times for Brownian motion and random walks in two dimensions
成果类型：
Article
署名作者：
Dembo, A; Peres, Y; Rosen, J; Zeitouni, O
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2004.160.433
发表日期：
2004
页码：
433-464
关键词：
摘要：
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.