Dimension of the SLE light cone, the SLE fan, and SLE$_κ(ρ)$ for $κ\in (0,4)$ and $ρ\in [\tfracκ{2}-4,-2)$

Jason Miller

Published 2016-06-22Version 1

Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $\kappa \in (0,4)$. The SLE$_\kappa$ light cone ${\mathbf L}(\theta)$ of $h$ with opening angle $\theta \in [0,\pi]$ is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field $e^{i h/\chi}$, $\chi = \tfrac{2}{\sqrt{\kappa}} - \tfrac{\sqrt{\kappa}}{2}$, with angles in $[-\tfrac{\theta}{2},\tfrac{\theta}{2}]$. We derive the Hausdorff dimension of ${\mathbf L}(\theta)$. If $\theta =0$ then ${\mathbf L}(\theta)$ is an ordinary SLE$_{\kappa}$ curve (with $\kappa < 4$); if $\theta = \pi$ then ${\mathbf L}(\theta)$ is the range of an SLE$_{\kappa'}$ curve ($\kappa' = 16/\kappa > 4$). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE. We also consider SLE$_\kappa(\rho)$ processes, which were originally only defined for $\rho > -2$, but which can also be defined for $\rho \leq -2$ using L\'evy compensation. The range of an SLE$_\kappa(\rho)$ is qualitatively different when $\rho \leq -2$. In particular, these curves are self-intersecting for $\kappa < 4$ and double points are dense, while ordinary SLE$_\kappa$ is simple. It was previously shown (Miller-Sheffield, 2016) that certain SLE$_\kappa(\rho)$ curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLE$_\kappa(\rho)$ for all values of $\rho$. Finally, we show that the Hausdorff dimension of the so-called SLE$_\kappa$ fan is the same as that of ordinary SLE$_\kappa$.