Imaginary geometry II: Reversibility of $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ for $\kappa\in(0,4)$

Abstract

Given a simply connected planar domain $D$, distinct points $x,y\in\partial D$, and $\kappa>0$, the Schramm–Loewner evolution $\operatorname{SLE}_{\kappa}$ is a random continuous non-self-crossing path in $\overline{D}$ from $x$ to $y$. The $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ processes, defined for $\rho_{1},\rho_{2}>-2$, are in some sense the most natural generalizations of $\operatorname{SLE}_{\kappa}$.

When $\kappa\leq4$, we prove that the law of the time-reversal of an $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ from $x$ to $y$ is, up to parameterization, an $\operatorname{SLE}_{\kappa}(\rho_{2};\rho_{1})$ from $y$ to $x$. This assumes that the “force points” used to define $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ are immediately to the left and right of the $\operatorname{SLE}$ seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit $\partial D\setminus\{x,y\}$.

The proof of time-reversal symmetry makes use of the interpretation of $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ as a ray of a random geometry associated to the Gaussian-free field. Within this framework, the time-reversal result allows us to couple two instances of the Gaussian-free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.