Geometric Interpretation of Half-Plane Capacity
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Lalley-2009-Geometric interpretation.pdf
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104.27 KB
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Checksum (MD5)
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Author(s)
Lalley, Steven P.
Lawler, Gregory F.
Narayanan, Hariharan
Date Issued
December 2009
Journal
Electronic Communications in Probability
Publisher
Institute of Mathematical Statistics
Citation
Lalley, Steven P., Gregory F. Lawler, and Hariharan Narayanan. “Geometric Interpretation of Half-Plane Capacity.” Electronic Communications in Probability 14, no. 0 (January 1, 2009).
Version
Final published version
Abstract
Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and complex analytic properties and makes SLE a time-homogeneous Markov process on conformal maps. In this note, we show that the half-plane capacity of a hull A is comparable up to multiplicative constants to more geometric quantities, namely the area of the union of all balls centered in A tangent to R, and the (Euclidean) area of a 1-neighborhood of A with respect to the hyperbolic metric.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1214/ECP.v14-1517
