Electronic Communications in Probability

Electronic Communications in Probability > Vol. 14 (2009) > Paper 55

Geometric Interpretation of Half-Plane Capacity

Steven P. Lalley, University of Chicago
Gregory F. Lawler,
Hariharan Narayanan, MIT

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.

Full text: PDF

Pages: 566-571

Published on: December 21, 2009

Bibliography

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 MR2129588 (2006i:60003)

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Electronic Communications in Probability. ISSN: 1083-589X