Cover Times for Brownian Motion and Random Walks in Two Dimensions
| dc.creator | Dembo, Amir | |
| dc.creator | Peres, Yuval | |
| dc.creator | Rosen, Jay | |
| dc.creator | Zeitouni, Ofer | |
| dc.date | 2001-07-26 | |
| dc.date | 2003-11-27 | |
| dc.date.accessioned | 2026-07-07T04:42:44Z | |
| dc.date.available | 2026-07-07T04:42:44Z | |
| dc.description | 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 sup_{x} 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 Z_n^2 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 Z^2 to cover the disc of radius n. | |
| dc.description | Conflict between definitions of constants in section 5 cleared on November 26, 2003 | |
| dc.identifier | https://arxiv.org/abs/math/0107191 | |
| dc.identifier | http://arxiv.org/abs/math/0107191 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61911 | |
| dc.subject | Probability | |
| dc.subject | Combinatorics | |
| dc.subject | Differential Geometry | |
| dc.title | Cover Times for Brownian Motion and Random Walks in Two Dimensions | |
| dc.type | text |