tailieunhanh - Đề tài " Basic properties of SLE "

SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. 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 κ = 8 the SLE trace is a path; for κ ∈ [0, 4] it is a simple path; for κ ∈ (4, 8) it is. | Annals of Mathematics Basic properties of SLE By Steffen Rohde and Oded Schramm Annals of Mathematics 161 2005 883 924 Basic properties of SLE By Steffen Rohde and Oded Schramm Dedicated to Christian Pommerenke on the occasion of his 70th birthday Abstract SLEk is a random growth process based on Loewner s equation with driving parameter a one-dimensional Brownian motion running with speed K. 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 K 8 the SLE trace is a path for K E 0 4 it is a simple path for K E 4 8 it is a self-intersecting path and for K 8 it is space-filling. It is also shown that the Hausdorff dimension of the SLEk trace is almost surely . at most 1 k 8 and that the expected number of disks of size e needed to cover it inside a bounded set is at least e_ 1 K 8 o 1 for K E 0 8 along some sequence e 0. Similarly for K 4 the Hausdorff dimension of the outer boundary of the SLEk hull is . at most 1 2 k and the expected number of disks of radius e needed to cover it is at least e_ 1 2 K o 1 for a sequence e 0. 1. Introduction Stochastic Loewner Evolution SLE is a random process of growth of a set Kt. The evolution of the set over time is described through the normalized conformal map gt gt z from the complement of Kt. The map gt is the solution of Loewner s differential equation with driving parameter a onedimensional Brownian motion. SLE or SLEk has one parameter K 0 which is the speed of the Brownian motion. A more complete definition appears in Section 2 below. The SLE process was introduced in Sch00 . There it was shown that under the assumption of the existence and conformal invariance of the scaling limit of loop-erased random walk the scaling limit is SLE2. See Figure . It