## Electronic Communications in Probability

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

### Geometric Interpretation of Half-Plane Capacity

Steven Lalley, Gregory Lawler, and Hariharan Narayanan

#### 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.

#### Article information

**Source**

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

**Dates**

Accepted: 21 December 2009

First available in Project Euclid: 6 June 2016

**Permanent link to this document**

http://projecteuclid.org/euclid.ecp/1465234764

**Digital Object Identifier**

doi:10.1214/ECP.v14-1517

**Mathematical Reviews number (MathSciNet)**

MR2576752

**Zentralblatt MATH identifier**

1191.60094

**Subjects**

Primary: 82B31: Stochastic methods

**Keywords**

Brownian motion Schramm-Loewner Evolution

**Rights**

This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

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.