Abstract
SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.
The present paper attempts a first systematic study of SLE. It is proved that for all κ ≠ 8 the SLE trace is a path; for κ ∈ [0, 4] it is a simple path; for κ ∈ (4, 8) it is a self-intersecting path; and for κ > 8 it is space-filling.
It is also shown that the Hausdorff dimension of the SLE κ trace is almost surely (a.s.) at most 1 + κ/8 and that the expected number of disks of size ε needed to cover it inside a bounded set is at least ε –(1 + κ/8 + o(1) for κ ∈ [0, 8) along some sequence \(\varepsilon \searrow 0\). Similarly, for ĸ ≥ 4, the Hausdorff dimension of the outer boundary of the SLE κ hull is a.s. at most 1 + 2/κ, and the expected number of disks of radius ε needed to cover it is at least ε –(1 + 2/κ) + o (1) for a sequence \(\varepsilon \searrow 0\).
Dedicated to Christian Pommerenke on the occasion of his 70th birthday
*Partially supported by NSF Grants DMS-0201435 and DMS-0244408.
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Acknowledgments
The authors thank Richard Kenyon for useful discussions and Jeff Steiff for numerous comments on a previous version of this paper.
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Rohde*, S., Schramm, O. (2011). Basic properties of SLE. In: Benjamini, I., Häggström, O. (eds) Selected Works of Oded Schramm. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9675-6_31
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