Basic properties of SLE

Steffen Rohde, Oded Schramm

Published 2001-06-05, updated 2004-01-20Version 4

SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional 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 self-intersecting path; and for $\kappa>8$ it is space-filling. 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 $\eps\to 0$.