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

Abstract

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 $\mathrm{SLE}_{\kappa }$. 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 $[-\theta /2,\theta /2]$. It is known that when $\theta =0$, the light cone looks like $\mathrm{SLE}_{\kappa }$, and when $\theta =\pi $ it looks like the range of an $\mathrm{SLE}_{16/\kappa }$ counterflow line. 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 agrees in law with the range of an $\mathrm{SLE}_{\kappa }(\rho )$ process with $\rho \in ((-2-\kappa /2)\vee (\kappa /2-4),-2)$. Conversely, the range of any such $\mathrm{SLE}_{\kappa }(\rho )$ process agrees in law with a non-space-filling light cone. As a consequence of our analysis, we obtain the first proof that these $\mathrm{SLE}_{\kappa }(\rho )$ processes are a.s. continuous curves and show that they can be constructed as natural path-valued functions of the GFF.