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2009 Geometric Interpretation of Half-Plane Capacity
Steven Lalley, Gregory Lawler, Hariharan Narayanan
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Electron. Commun. Probab. 14: 566-571 (2009). DOI: 10.1214/ECP.v14-1517


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.


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Steven Lalley. Gregory Lawler. Hariharan Narayanan. "Geometric Interpretation of Half-Plane Capacity." Electron. Commun. Probab. 14 566 - 571, 2009.


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

zbMATH: 1191.60094
MathSciNet: MR2576752
Digital Object Identifier: 10.1214/ECP.v14-1517

Primary: 82B31

Keywords: Brownian motion , Schramm-Loewner evolution

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