SLE$_{\kappa}(\rho)$ processes in the light cone regime on Liouville quantum gravity

Abstract

We study the relationship between certain SLE$_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution with parameter $\kappa$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $\rho > -2-\kappa/2$ and in this work we will focus on the light cone regime, meaning that $\kappa \in (0,4)$ and $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$. Such processes are self-intersecting even though ordinary SLE$_\kappa$ curves are simple for $\kappa \in (0,4)$. We show that such a process drawn on top of an independent $\sqrt{\kappa}$-LQG surface called a weight $(\rho+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $\alpha$-stable L\'evy process with $\alpha = 1-2(\rho+2)/\kappa$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_\kappa(\kappa-4)$ curve on an independent $\sqrt{\kappa}$-LQG surface for $\kappa \in (4/3,2)$.