Spacefilling Curves and Phases of the Loewner Equation
Complex Variables
2011-03-02 v1 Probability
Abstract
Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C>4 such that a Loewner driving term whose trace is space filling has Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpinski gasket fall into this class.
Cite
@article{arxiv.1103.0071,
title = {Spacefilling Curves and Phases of the Loewner Equation},
author = {Joan Lind and Steffen Rohde},
journal= {arXiv preprint arXiv:1103.0071},
year = {2011}
}
Comments
18 pages, 11 figures