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SLE as a Mating of Trees in Euclidean Geometry
Name
220_2018_3149_ReferencePDF.pdf
Size
22.97 MB
Format
Adobe PDF
Checksum (MD5)
61214ad06386d3cb9cb04a6a75c45750
Author(s)
Holden, Nina
Sun, Xin
Date Issued
May 15, 2018
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
The mating of trees approach to Schramm–Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014.
arXiv:1409.7055
). In this paper we consider the mating of trees approach to SLE in Euclidean geometry. Let
$${\eta}$$
η
be a whole-plane space-filling SLE with parameter
$${\kappa > 4}$$
κ
>
4
, parameterized by Lebesgue measure. The main observable in the mating of trees approach is the contour function, a two-dimensional continuous process describing the evolution of the Minkowski content of the left and right frontier of
$${\eta}$$
η
. We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve
$${\eta}$$
η
. We also prove that the uniform spanning tree on
$${\mathbb{Z}^2}$$
Z
2
converges to SLE8 in the natural topology associated with the mating of trees approach.
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DOI of Published Version
https://doi.org/10.1007/s00220-018-3149-1
