Values of Brownian intersection exponents II: Plane exponents
| dc.creator | Lawler, Gregory F. | |
| dc.creator | Schramm, Oded | |
| dc.creator | Werner, Wendelin | |
| dc.date | 2000-03-27 | |
| dc.date | 2000-12-13 | |
| dc.date.accessioned | 2026-07-07T10:22:29Z | |
| dc.date.available | 2026-07-07T10:22:29Z | |
| dc.description | We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0,t] and B'[0,t] do not intersect decays like t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)| =1, for all t \ge 1, c^{-1} t^{-5/8} \le ¶[ B[0,t] \cap B'[0,t] = \emptyset ] \le c t^{-5/8}. One consequence is that the set of cut-points of B[0,1] has Hausdorff dimension 3/4 almost surely. The values of other exponents are also derived. Using an analyticity result, which is to be established in a forthcoming paper, it follows that the Hausdorff dimension of the outer boundary of B[0,1] is 4/3, as conjectured by Mandelbrot. The proofs are based on a study of SLE_6 (stochastic Loewner evolution with parameter 6), a recently discovered process which conjecturally is the scaling limit of critical percolation cluster boundaries. The exponents of SLE_6 are calculated, and they agree with the physicists' predictions for the exponents for critical percolation and self-avoiding walks. From the SLE_6 exponents the Brownian intersection exponents are then derived. | |
| dc.description | To appear in Acta Mathematica | |
| dc.identifier | https://arxiv.org/abs/math/0003156 | |
| dc.identifier | http://arxiv.org/abs/math/0003156 | |
| dc.identifier | ActaMath.187:275-308,2001 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/175529 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60J65; 30C35; 82B41; 82B43 | |
| dc.title | Values of Brownian intersection exponents II: Plane exponents | |
| dc.type | text |