Abstract
Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix \({\kappa \in (0,4)}\). The \({{\rm SLE}_\kappa}\) light cone 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 \({e^{ih/\chi}}\), \({\chi = \tfrac{2}{\sqrt{\kappa}} - \tfrac{\sqrt{\kappa}}{2}}\), with angles in \({[-\tfrac{\theta}{2},\tfrac{\theta}{2}]}\). We derive the Hausdorff dimension of L\({(\theta)}\).
If \({\theta =0}\) then L\({(\theta)}\) is an ordinary \({{\rm SLE}_{\kappa}}\) curve (with \({\kappa < 4}\)); if \({\theta = \pi}\) then L\({(\theta)}\) is the range of an \({{\rm SLE}_{\kappa'}}\) curve (\({\kappa' = 16/\kappa > 4}\)). In these extremes, this leads to a new proof of the Hausdorff dimension formula for \({{\rm SLE}}\).
We also consider \({{\rm SLE}_\kappa(\rho)}\) processes, which were originally only defined for \({\rho > -\,2}\), but which can also be defined for \({\rho \leq -2}\) using Lévy compensation. The range of an \({{\rm SLE}_\kappa(\rho)}\) is qualitatively different when \({\rho \leq -2}\). In particular, these curves are self-intersecting for \({\kappa < 4}\) and double points are dense, while ordinary \({{\rm SLE}_\kappa}\) is simple. It was previously shown (Miller and Sheffield in Gaussian free field light cones and \({{\rm SLE}_\kappa(\rho)}\), 2016) that certain \({{\rm SLE}_\kappa(\rho)}\) curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of \({{\rm SLE}_\kappa(\rho)}\) for all values of \({\rho}\).
Finally, we show that the Hausdorff dimension of the so-called \({{\rm SLE}_\kappa}\) fan is the same as that of ordinary \({{\rm SLE}_\kappa}\).
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Communicated by H. Duminil-Copin
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Miller, J. Dimension of the SLE Light Cone, the SLE Fan, and \({{\rm SLE}_\kappa(\rho)}\) for \({\kappa \in (0,4)}\) and \({\rho \in}\) \({\big[{\tfrac{\kappa}{2}}-4,-2\big)}\). Commun. Math. Phys. 360, 1083–1119 (2018). https://doi.org/10.1007/s00220-018-3109-9
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DOI: https://doi.org/10.1007/s00220-018-3109-9