Abstract

On the continuity of SLE curves in k and their behavior at the tip

The Schramm-Loewner evolution with parameter k > 0, SLE , is a family of random fractal curves constructed using the Loewner differential equation driven by a standard Brownian motion times the square-root of k. These curves arise as scaling limits of cluster interfaces in certain planar critical lattice models. A natural question that has been asked is whether the (parameterized) SLE curves almost surely change continuously if the Brownian motion sample is kept fixed while k is varied.

In the talk I will present recent work giving a positive answer to this question, at least for an interval of k. I will also give some background on SLE and the Loewner equation, and describe the basic tools we use for the proof, in particular certain bounds characterizing the behavior of a growing SLE curve close to its tip.

The talk is based on joint work with Steffen Rohde and Carto Wong, and Greg Lawler.