Duality of chordal SLE

By Dapeng Zhan. 2007

97 citations recorded by the source corpus.

We derive some geometric properties of chordal SLE(κ;$\vec{\rho}$) processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE(κ;$\vec{\rho}$) process has the same distribution as the image of a chordal SLE(κ’;$\vec{\rho}$’) trace, where κ>4, κ’=16/κ, and the forces $\vec{\rho}$ and $\vec{\rho}$’ are suitably chosen. We find that for κ≥8, the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed $x\in\mathbb{R}\setminus\{0\}$ is the image of some SLE(16/κ;$\vec{\rho}$) trace started from x. Then we obtain a new proof of the fact that chordal SLE(κ) trace is not reversible for κ>8. We also prove that the reversal of SLE(4;$\vec{\rho}$) trace has the same distribution as the time-change of some SLE(4;$\vec{\rho}$’) trace for certain values of $\vec{\rho}$ and $\vec{\rho}$’.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7c7e0d93b9034e25c7aa3319ed3e35e1e4a4db0c

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Citations in the corpus

Imaginary geometry I: interacting SLEs (2012, result): “This was proven in certain special cases by Zhan [Zha08, Zha10] and Dubédat [Dub09a].”
SLE and the free field: partition functions and couplings (2007, background): “Various such identities are established in [10, 41].”
Reversibility of chordal SLE (2008, methodology): “It may also be used to study the reversal of the trace of other variations of SLE, for example, SLE(κ,ρ) [5], continuous LERW [14] and annulus SLE [15, 16].”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ ∈ [0, 4]) to all κ ∈ [0, 8].”