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arXiv · 1512.01223

An almost sure KPZ relation for SLE and Brownian motion

Ewain Gwynne·Nina Holden·Jason Miller
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Abstract

The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well-known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE$_κ$ curve for $κ\not=4$; the double points and cut points of SLE$_κ$ for $κ>4$; and the intersection of two flow lines of a Gaussian free field. We also obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE$_κ$ for $κ>4$ and $m \geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $π/2$-cone times of a correlated planar Brownian motion.

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BibTeXRIS

Ewain Gwynne, Nina Holden, Jason Miller. 2019-07-02. An almost sure KPZ relation for SLE and Brownian motion. https://arxiv.org/abs/1512.01223

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