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

TL;DR

This paper explores the properties of SLE$_{ ho}$ processes in the light cone regime on Liouville quantum gravity surfaces, revealing their representation as glued trees related to correlated stable Lévy processes and connecting them to bipolar oriented maps.

Contribution

It introduces a new representation of SLE$_{ ho}$ processes in the light cone regime as glued trees described by correlated stable Lévy processes, linking them to quantum gravity surfaces and planar maps.

Findings

SLE$_{ ho}$ processes can be represented as glued trees on LQG surfaces.

The processes are described by a correlated $ ext{α}$-stable Lévy process with $ ext{α} = 1 - 2( ho+2)/ ext{κ}$.

Bipolar oriented maps with large faces relate to SLE$_{ ext{κ}}( ext{κ}-4)$ on LQG surfaces for $ ext{κ} ext{ in } (4/3,2)$.

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 $\kappa \in (4/3,2)$.