ELECTRONIC COMMUNICATIONS in PROBABILITY GEOMETRIC INTERPRETATION OF HALF-PLANE CAPACITY

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.


Introduction
Suppose A is a bounded, relatively closed subset of the upper half plane . We call A a compacthull if A is bounded and \ A is simply connected. The half-plane capacity of A, hcap(A), is defined in a number of equivalent ways (see [1], especially Chapter 3). If g A denotes the unique conformal transformation of \ A onto with g A (z) = z + o(1) as z → ∞, then g A has the expansion Equivalently, if B t is a standard complex Brownian motion and τ A = inf{t ≥ 0 : B t ∈ \ A}, Let Im[A] = sup{Im(z) : z ∈ A}. Then if y ≥ Im[A], we can also write These last two definitions do not require \ A to be simply connected, and the latter definition does not require A to be bounded but only that Im[A] < ∞.
For -hulls (that is, for relatively closed A for which \ A is simply connected), the half-plane capacity is comparable to a more geometric quantity that we define. This is not new (the second author learned it from Oded Schramm in oral communication), but we do not know of a proof in the literature 3 . In this note, we prove the fact giving (nonoptimal) bounds on the constant. We start with the definition of the geometric quantity.
where (z, ε) denotes the disk of radius ε about z.
In this paper, we prove the following.

Proof of Theorem 1
It suffices to prove this for weakly bounded -hulls, by which we mean -hulls A with Im(A) < ∞ and such that for each ε > 0, the set {x + i y : y > ε} is bounded. Indeed, for -hulls that are not weakly bounded, it is easy to verify that hsiz(A) = hcap(A) = ∞.
We start with a simple inequality that is implied but not explicitly stated in [1]. Equality is achieved when A is a vertical line segment.

Lemma 1. If A is an -hull, then
Proof. Due to the continuity of hcap with respect to the Hausdorff metric on -hulls, it suffices to prove the result for -hulls that are path-connected. For two -hulls A 1 ⊆ A 2 , it can be seen using the Optional stopping theorem that hcap(A 1 ) ≤ hcap(A 2 ). Therefore without loss of generality, A can be assumed to be of the form η(0, T ] where η is a simple curve with η(0+) ∈ , parameterized so that hcap[η(0, t]) = 2t. In particular, T = hcap(A)/2. If g t = g η(0,t] , then g t satisfies the Loewner equation where U : The next lemma is a variant of the Vitali covering lemma. If c > 0 and z = x + i y ∈ , let Lemma 2. Suppose A is a weakly bounded -hull and c > 0. Then there exists a finite or countably infinite sequence of points {z 1 = x i + i y 1 , z 2 = x 2 + i y 2 , , . . .} ⊂ A such that: • y 1 ≥ y 2 ≥ y 3 ≥ · · · ; • the intervals (x 1 , c), (x 2 , c), . . . are disjoint; Proof. We define the points recursively. Let A 0 = A and given {z 1 , . . . , z j }, let Using the weak boundedness of A, we can see that y j → 0 and hence (3) holds.
Lemma 3. For every c > 0, let Then, for any c > 0, if A is a weakly bounded -hull and x 0 + i y 0 ∈ A with y 0 = Im(A), then Proof. By scaling and invariance under real translation, we may assume that Im[A] = y 0 = 1 and x 0 = 0. Let S = S c be defined to be the set of all points z of the form x + iu y where x + i y ∈ A \ (i, 2c) and 0 < u ≤ 1.
Using the capacity inequality [1, (3.10)] we see that Hence, it suffices to show that By properties of halfplane capacity [1, (3.8)] and (1), Hence, it suffices to prove that , The map Φ(z) = sin (θ z) maps V onto sending [−2c, 2c] to [−1, 1] and Φ(i) = i sinh θ . Using conformal invariance of Brownian motion and the Poisson kernel in , we see that The second equality uses the double angle formula for the tangent.