منابع مشابه
Geometric Interpretation of Half-Plane Capacity
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 com...
متن کاملA 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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Let G be an infinite geometric graph; in particular, a graph whose vertices are a countable discrete set of points on the plane, with vertices u, v adjacent if their Euclidean distance is less than 1. A “fire” begins at some finite set of vertices and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time-step. Let f(G) be the least f for which any fi...
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Let Gλ be the graph whose vertices are points of a planar Poisson process of density λ, with vertices adjacent if they are within distance 1. A “fire” begins at some vertex and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time-step. Let fλ be the least f such that, with probability 1, any fire on Gλ can be stopped in finite time. We show that fλ ...
متن کاملOn the Linear Combinations of Slanted Half-Plane Harmonic Mappings
In this paper, the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.
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ژورنال
عنوان ژورنال: Computational Geometry
سال: 2021
ISSN: 0925-7721
DOI: 10.1016/j.comgeo.2020.101728