Imaginary geometry III: reversibility of SLE_\kappa\ for \kappa \in (4,8)

TL;DR

This paper proves the reversibility of SLE_rac{} in the range (4,8), establishing key properties of these curves and their relation to conformal loop ensembles and Gaussian free fields.

Contribution

It demonstrates the reversibility of SLE_rac{} processes for rac{} in (4,8) and characterizes when SLE_rac{}( ho_1; ho_2) processes are reversible, filling a gap in the theory.

Findings

Reversibility of SLE_rac{} for rac{} in (4,8) is established.

SLE_rac{}( ho_1; ho_2) processes are reversible if and only if both rac{} are at least rac{}/2-4.

The result enables the canonical definition of CLE_rac{} with continuous loops.

Abstract

Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x to y in D. We prove that the law of the time-reversal of \eta is, up to reparameterization, an SLE_\kappa process from y to x in D. More generally, we prove that SLE_\kappa(\rho_1;\rho_2) processes are reversible if and only if both \rho_i are at least \kappa/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all \kappa \in (4,8) the so-called conformal loop ensembles CLE_\kappa\ are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are SLE_\kappa curves.