Strong path convergence from Loewner driving function convergence

Abstract

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal ${\mathrm{SLE}}_{\kappa }$ as a scaling limit, with $\kappa <8$ (nonspace-filling). Existing ${\mathrm{SLE}}_{\kappa }$ convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from $\infty$) converge to Brownian motion. Unfortunately, this is not sufficient, and additional arguments are required to complete the proofs. We show that driving function convergence is sufficient if it can be established for both parametrization directions and a generic observation point.

Published in at http://dx.doi.org/10.1214/10-AOP627 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Critical percolation in the plane
• Reversibility of chordal SLE
• Universality in the 2D Ising model and conformal invariance of fermionic observables
• Towards conformal invariance of 2D lattice models

Cited by in corpus (5)

• Universality for SLE(4)
• A contour line of the continuum Gaussian free field
• A deterministic approach to Loewner-energy minimizers
• A level line of the Gaussian free field with measure-valued boundary conditions
• Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice