On the Continuity of SLE(\kappa) in \kappa

TL;DR

This paper proves that Schramm-Loewner Evolution (SLE) curves depend continuously on the parameter  in a specific interval, providing estimates for their modulus of continuity and the change in curves as  varies.

Contribution

It establishes the almost sure continuity of SLE() curves with respect to  in a certain range and provides quantitative estimates for this dependence.

Findings

SLE() curves are continuous in  within [0, _0) for almost every Brownian sample.

The paper provides explicit estimates for the modulus of continuity of the curves.

It quantifies how the curves change as  varies, with bounds depending on .

Abstract

We prove that for almost every Brownian motion sample, the corresponding SLE(\kappa) curves parameterized by capacity exist and change continuously in the supremum norm when \kappa varies in the interval [0,\kappa_0), where \kappa_0=8(2-\sqrt{3})=2.143... We estimate the \kappa-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with \kappa.