2001/07/26 by Amir Dembo, Yuval Peres, Dembo, Amir +5 · 1 citation
Mathematics · #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR) #math.CO #math.DG #math.PR
paper · pdf · doi:10.48550/arxiv.math/0107191
Conflict between definitions of constants in section 5 cleared on November 26, 2003
arxiv created 2003/11/27 · arxiv updated 2009/11/30
Let T(x,r) denote the first hitting time of the disc of radius r centered at x for Brownian motion on the two dimensional torus. We prove that supx T(x,r)/|log r|2 --> 2/pi as r --> 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 (2n log n)2/pi. 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 Revesz, that describes the asymptotics for the number of steps needed by simple random walk in Z2 to cover the disc of radius n.