Imaginary geometry II: reversibility of SLE_\kappa(\rho_1;\rho_2) for \kappa \in (0,4)

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Abstract

Given a simply connected planar domain D, distinct points x,y \in \partial D, and \kappa >0, the Schramm-Loewner evolution SLE_\kappa is a random continuous non-self-crossing path in the closure of D from x to y. The SLE_\kappa(\rho_1;\rho_2) processes, defined for \rho_1, \rho_2 > -2, are in some sense the most natural generalizations of SLE_\kappa. When \kappa \leq 4, we prove that the law of the time-reversal of an \SLE_\kappa(\rho_1;\rho_2) from x to y is, up to parameterization, an SLE_\kappa(\rho_2;\rho_1) from y to x. This assumes that the "force points" used to define SLE_\kappa(\rho_1;\rho_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 the boundary of D except at the endpoints. The time-reversal symmetry has a particularly natural interpretation when the paths are coupled with the Gaussian free field and viewed as rays of a random geometry. It 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.