Dimension of the SLE Light Cone, the SLE Fan, and $${{\rm SLE}_\kappa(\rho)}$$ SLE κ ( ρ ) for $${\kappa \in (0,4)}$$ κ ∈ ( 0 , 4 ) and $${\rho \in}$$ ρ ∈ $${\big[{\tfrac{\kappa}{2}}-4,-2\big)}$$ [ κ 2 - 4 , - 2 )

07 Mar 2018

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