Gaussian free field light cones and SLE$_\kappa(\rho)$

Abstract

We derive a surprising correspondence between SLE$_{\kappa}(\rho)$ processes and light cones of the Gaussian free field (GFF). Recall that (one-sided, chordal, origin-seeded) SLE$_\kappa(\rho)$ processes are in some sense the simplest and most natural variants of the Schramm-Loewner evolution. They were originally defined only for $\rho > -2$, but one can use L\'evy compensation to extend the definition to any $\rho > -2-\tfrac{\kappa}{2}$ and to obtain qualitatively different curves. The triangle $T = \{(\kappa, \rho): (-2-\tfrac{\kappa}{2})\vee (\tfrac{\kappa}{2}-4) < \rho < -2 \}$ is the primary focus of this paper. When $(\kappa, \rho) \in T$, the SLE$_\kappa(\rho)$ curves are highly non-simple (and double points are dense) even though $\kappa < 4$. Let $h$ be an instance of the GFF. Fix $\kappa\in (0,4)$ and $\chi = 2/\sqrt{\kappa} - \sqrt{\kappa}/2$. Recall that an imaginary geometry ray is a flow line of $e^{i(h/\chi +\theta)}$ that looks locally like SLE$_\kappa$. The light cone with parameter $\theta \in [0, \pi]$ is the set of points reachable from the origin by a sequence of rays with angles in $[-\theta/2, \theta/2]$. When $\theta=0$, the light cone looks like SLE$_\kappa$, and when $\theta = \pi$ it looks like the range of an SLE$_{16/\kappa}$. We find that when $\theta \in (0, \pi)$ the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every non-space-filling light cone (with $\theta \in (0,\pi]$ and $\kappa \in (0,4)$) agrees in law with the range of an SLE$_\kappa(\rho)$ process with $(\kappa, \rho) \in T$. Conversely, the range of any SLE$_\kappa(\rho)$ with $(\kappa,\rho) \in T$ agrees in law with a non-space-filling light cone. As a consequence, we obtain the first proof that these SLE$_\kappa(\rho)$ processes are continuous and show that they are natural path-valued functions of the GFF.