Boundary proximity of SLE
| dc.creator | Schramm, Oded | |
| dc.creator | Zhou, Wang | |
| dc.date | 2007-11-21 | |
| dc.date | 2007-12-06 | |
| dc.date.accessioned | 2026-07-07T08:47:22Z | |
| dc.date.available | 2026-07-07T08:47:22Z | |
| dc.description | This paper examines how close the chordal $\SLE_κ$ curve gets to the real line asymptotically far away from its starting point. In particular, when $κ\in(0,4)$, it is shown that if $β>β_κ:=1/(8/κ-2)$, then the intersection of the $\SLE_κ$ curve with the graph of the function $y=x/(\log x)^β$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $β=β_κ$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^α}$, $x>e^e$, in an unbounded set if $α\le 1$, but not if $α>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_κ$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_κ$ curve and real axis is $2-8/κ$ when $4<κ<8$. | |
| dc.description | 18 pages, new results are added, typos are corrected | |
| dc.identifier | https://arxiv.org/abs/0711.3350 | |
| dc.identifier | http://arxiv.org/abs/0711.3350 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143574 | |
| dc.subject | Probability | |
| dc.subject | Complex Variables | |
| dc.subject | 60D05, 28A80 | |
| dc.title | Boundary proximity of SLE | |
| dc.type | text |