Intersection of SLE Paths: the double and cut point dimension of SLE

Hao Wu Yau Mathematical Science Center, Tsinghua University, China Jason Miller Department of Pure Mathematics and Mathematics Statistics, University of Cambridge, UK

Probability mathscidoc:1707.28001

Probab. Theorey Relat. Fields, (167), 45-105, 2017.1
We compute the almost-sure Hausdorff dimension of the double points of chordal SLE$_\kappa$ for $\kappa > 4$, confirming a prediction of Duplantier--Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE$_\kappa$ for $\kappa > 4$ as well as analogous dimensions for the radial and whole-plane SLE$_\kappa(\rho)$ processes for $\kappa > 0$. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field $e^{ih/\chi}$, where $h$ is a Gaussian free field and $\chi > 0$, of different angles with each other and with the domain boundary.
Schramm Loewner Evolution (SLE), Hausdorff dimension, double points, cut points, Gaussian Free Field (GFF), Imaginary Geometry
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@inproceedings{hao2017intersection,
  title={Intersection of SLE Paths: the double and cut point dimension of SLE},
  author={Hao Wu, and Jason Miller},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170717102624338619794},
  booktitle={Probab. Theorey Relat. Fields},
  number={167},
  pages={45-105},
  year={2017},
}
Hao Wu, and Jason Miller. Intersection of SLE Paths: the double and cut point dimension of SLE. 2017. In Probab. Theorey Relat. Fields. pp.45-105. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170717102624338619794.
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