Basic properties of SLE

SLE kappa 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 not equal 8 the SLE trace is a path; for kappa is an element of [0, 4] it is a simple path; for kappa is an element of (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 almost surely (a.s.) at most 1 + kappa/8 and that the expected number of disks of size E needed to cover it inside a bounded set is at least epsilon(-(1+kappa/8)+o(1)) for kappa is an element of [0, 8) along some sequence epsilon SE arrow 0. Similarly, for K > 4, the Hausdorff dimension of the outer boundary of the SLE kappa, hull is a.s. at most 1 + 2/kappa, and the expected number of disks of radius epsilon needed to cover it is at least epsilon(-(1+2/kappa)+o(1)) for a sequence E SE arrow 0.