Geometric Interpretation of Half-Plane Capacity
نویسندگان
چکیده
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.
منابع مشابه
A Geometric Interpretation of Half-Plane Capacity
Let A be a bounded, relatively closed subset of the upper half plane H whose complement in H is simply connected. If Bt is a standard complex Brownian motion and τA = inf{t ≥ 0 : Bt 6∈ H \A}, the half-plane capacity hcap(A) is defined as hcap(A) := lim y→∞ y E [Im(BτA)] . This quantity arises in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a...
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