Basic properties of SLE
Abstract
SLE is a random growth process based on Loewner's equation with driving parameter a onedimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $\kappa\ne 8$ the SLE trace is a path; for $\kappa\in[0,4]$ it is a simple path; for $\kappa\in(4,8)$ it is a selfintersecting path; and for $\kappa>8$ it is spacefilling. It is also shown that the Hausdorff dimension of the SLE trace is a.s. at most $1+\kappa/8$ and that the expected number of disks of size $\eps$ needed to cover it inside a bounded set is at least $\eps^{(1+\kappa/8)+o(1)}$ for $\kappa\in[0,8)$ along some sequence $\eps\to 0$. Similarly, for $\kappa\ge 4$, the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most $1+2/\kappa$, and the expected number of disks of radius $\eps$ needed to cover it is at least $\eps^{(1+2/\kappa)+o(1)}$ for a sequence $\eps\to 0$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2001
 arXiv:
 arXiv:math/0106036
 Bibcode:
 2001math......6036R
 Keywords:

 Mathematics  Probability;
 Mathematics  Complex Variables;
 Mathematics  Mathematical Physics;
 Mathematical Physics;
 60G17;
 30C35;
 60K35;
 60J99
 EPrint:
 Made several corrections