Reversibility of whole-plane SLE for $\kappa > 8$

TL;DR

This paper proves that whole-plane SLE$_ppa$ curves are reversible for all ppa > 8, completing the classification and confirming a long-standing conjecture using a new mating-of-trees approach involving Liouville quantum gravity.

Contribution

It establishes the reversibility of whole-plane SLE for ppa > 8, extending previous results and introducing a novel mating-of-trees theorem.

Findings

Reversibility of whole-plane SLE for ppa > 8 proven.

Introduces a new mating-of-trees theorem involving Liouville quantum gravity.

Completes the classification of SLE reversibility across all ppa values.

Abstract

Whole-plane SLE$_\kappa$ is a random fractal curve between two points on the Riemann sphere. Zhan established for $\kappa \leq 4$ that whole-plane SLE$_\kappa$ is reversible, meaning invariant in law under conformal automorphisms swapping its endpoints. Miller and Sheffield extended this to $\kappa \leq 8$. We prove whole-plane SLE$_\kappa$ is reversible for $\kappa > 8$, resolving the final case and answering a conjecture of Viklund and Wang. Our argument depends on a novel mating-of-trees theorem of independent interest, where Liouville quantum gravity on the disk is decorated by an independent radial space-filling SLE curve.