Ergodicity of the tip of an SLE curve

TL;DR

This paper proves the reversibility of certain SLE curves for < and uses this to analyze the ergodic behavior of the curve near its tip, revealing new insights into SLE dynamics.

Contribution

It establishes reversibility of whole-plane SLE(,+2) and relates chordal and radial SLEs to this model, advancing understanding of SLE tip behavior.

Findings

Reversibility of whole-plane SLE(,+2) for <

Conformal mapping of chordal SLE to initial segments of whole-plane SLE

Ergodic properties of SLE tip points at fixed capacity time

Abstract

We first prove that, for $\kappa\in(0,4)$, a whole-plane SLE$(\kappa;\kappa+2)$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $\kappa\in(0,4)$, a chordal SLE$_\kappa$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(\kappa;\kappa+2)$ trace. A similar but weaker result holds for radial SLE$_\kappa$. These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE laminations and conformal removability of SLE$_\kappa$ curves for $\kappa\in(0,4)$.