Imaginary geometry II: Reversibility of SLE[subscript κ](ρ[subscript 1];ρ[subscript 2]) for κ ∈ (0,4)

Abstract

Given a simply connected planar domain D, distinct points x,y ∈ ∂D, and κ>0, the Schramm–Loewner evolution SLE[subscript κ] is a random continuous non-self-crossing path in [bar over D] from x to y. The SLE[subscript κ](ρ[subscript 1];ρ[subscript 2]) processes, defined for ρ[subscript 1],ρ[subscript 2] > −2, are in some sense the most natural generalizations of SLE[subscript κ]. When κ≤4, we prove that the law of the time-reversal of an SLE[subscript κ](ρ[subscript 1];ρ[subscript 2]) from x to y is, up to parameterization, an SLE[subscript κ](ρ[subscript 2];ρ[subscript 1])from y to x. This assumes that the “force points” used to define SLE[subscript κ](ρ[subscript 1];ρ[subscript 2]) are immediately to the left and right of the SLE seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit ∂D∖{x,y}.

The proof of time-reversal symmetry makes use of the interpretation of SLE[subscript κ](ρ[subscript 1];ρ[subscript 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.