Electronic Communications in Probability

Geometric Interpretation of Half-Plane Capacity

Steven Lalley, Gregory Lawler, and Hariharan Narayanan

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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.

Article information

Electron. Commun. Probab. Volume 14 (2009), paper no. 55, 566-571.

Accepted: 21 December 2009
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B31: Stochastic methods

Brownian motion Schramm-Loewner Evolution

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Lalley, Steven; Lawler, Gregory; Narayanan, Hariharan. Geometric Interpretation of Half-Plane Capacity. Electron. Commun. Probab. 14 (2009), paper no. 55, 566--571. doi:10.1214/ECP.v14-1517. http://projecteuclid.org/euclid.ecp/1465234764.

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  • Lawler, Gregory F. Conformally invariant processes in the plane.Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3