SLE coordinate changes

By Oded Schramm, David Bruce Wilson. 2005

180 citations recorded by the source corpus.

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ − 6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(κ; ρ1 ,...,ρ n) with respect to SLE(κ).

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/01058d4d7b8db8a15d2d9a006c8da65333185a45

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Citations in the corpus

Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “These variants are rather closely related to one another [83].”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “However, a variant of SLE(κ) called SLE(κ, κ−6) has been shown by Schramm and Wilson [38] (see also [39]) to be target-independent.”
A contour line of the continuum Gaussian free field (2010, background): “n parameterization. By Loewner’s equation, ∂tObt is a function of cW t−Obt. Therefore, the process cWt−Obt, together with cW 0 and 20 Ob 0 determines Obtand thus also Wct. This then determines Wt(see [SW05] for more details about changing between radial and chordal coordinates). 3 Local sets 3.1 Abs”
Game Theory, Alive (2017, background): “Nobel-prize winner William Vickrey (1960) discovered that one can combine the low communication cost of sealed-bid auctions with the simplicity of the optimal bidding rule in ascending auctions.”
Duality of chordal SLE (2007, background): “From the work in [14], we get the conformal invariance of chordal SLE(κ; ρ) processes, which is the following lemma.”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “We remark that one can consider radial SLEκ(ρ) processes with many boundary force points ρ as in [24], though we will not consider the more general case in this article.”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “By coordinate change ([26]) when coming back to the half-plane setting, the minimizing curve before hitting z can be viewed as a chordal SLE0(−8) starting at the origin, with inner marked point at z.”
Exploration trees and conformal loop ensembles (2006, background): “We will be particularly interested in chordal SLE(κ;κ − 6) because of the following [12]:”
Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, methodology): “ the literature: in this case, (4.6) takes the form dW t= Xn i=1 Re ˆ i f t(y i) dt+ p dB t; (4.7) which is the same expression one nds in the usual denition of the forwardow SLE ;ˆ process, as in [SW05]. (Note that the notation here diers from [SW05], since here we use ˆto denote the measure, not t”