# Hausdorff Dimension of the SLE Curve Intersected with the Real Line

@article{Alberts2007HausdorffDO, title={Hausdorff Dimension of the SLE Curve Intersected with the Real Line}, author={Tom Alberts and Scott Sheffield}, journal={Electronic Journal of Probability}, year={2007}, volume={13}, pages={1166-1188} }

We establish an upper bound on the asymptotic probability of an $SLE(\kappa)$ curve hitting two small intervals on the real line as the interval width goes to zero, for the range $4 < \kappa < 8$. As a consequence we are able to prove that the random set of points in $R$ hit by the curve has Hausdorff dimension $2-8/\kappa$, almost surely.

#### 28 Citations

The covariant measure of SLE on the boundary

- Mathematics
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We construct a natural measure μ supported on the intersection of a chordal SLE(κ) curve γ with $${\mathbb{R}}$$ , in the range 4 < κ < 8. The measure is a function of the SLE path in question.… Expand

Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line

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- 2015

The Schramm-Loewner evolution (SLE) is a probability measure on random fractal curves that arise as scaling limits of twodimensional statistical physics systems. In this paper we survey some results… Expand

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

- Mathematics, Physics
- 2013

We compute the almost-sure Hausdorff dimension of the double points of chordal $$\mathrm {SLE}_\kappa $$SLEκ for $$\kappa > 4$$κ>4, confirming a prediction of Duplantier–Saleur (1989) for the… Expand

A Dimension Spectrum for SLE Boundary Collisions

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We consider chordal SLE$${_\kappa}$$κ curves for $${\kappa > 4}$$κ>4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension $${{\rm… Expand

Boundary proximity of SLE

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This paper examines how close the chordal SLEκ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ϵ (0, 4), it is shown that if β > βκ := 1/(8/κ − 2),… Expand

Dimension transformation formula for conformal maps into the complement of an SLE curve

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We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $${\mathbb {R}}$$ R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to… Expand

An almost sure KPZ relation for SLE and Brownian motion

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The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\gamma$-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a… Expand

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- Mathematics
- 2007

We derive a number of estimates for the probability that a chordal SLE$_\kappa$ path in the upper half plane $\mathbb{H}$ intersects a semicircle centred on the real line. We prove that if $0 0$ of… Expand

Bridge Decomposition of Restriction Measures

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Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process,… Expand

Continuity of the SLE trace in simply connected domains

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We prove that the SLEκ trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of… Expand

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