Imaginary geometry II: Reversibility of SLEκ(ρ1;ρ2) for κ∈(0,4).
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Miller, Jason [1] ; Sheffield, Scott [2]
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[1] University of Cambridge
University of Cambridge
Cambridge District, Reino Unido
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[2] Massachusetts Institute of Technology
Massachusetts Institute of Technology
City of Cambridge, Estados Unidos
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 1647-1722
- Idioma: inglés
- Enlaces
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Resumen
Given a simply connected planar domain D, distinct points x,y∈∂D, and κ>0, the Schramm–Loewner evolution SLEκ is a random continuous non-self-crossing path in D¯¯¯¯ from x to y. The SLEκ(ρ1;ρ2) processes, defined for ρ1,ρ2>−2, are in some sense the most natural generalizations of SLEκ.
When κ≤4, we prove that the law of the time-reversal of an SLEκ(ρ1;ρ2) from x to y is, up to parameterization, an SLEκ(ρ2;ρ1) from y to x. This assumes that the “force points” used to define SLEκ(ρ1;ρ2) are immediately to the left and right of the SLE seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit ∂D∖{x,y}.
The proof of time-reversal symmetry makes use of the interpretation of SLEκ(ρ1;ρ2) as a ray of a random geometry associated to the Gaussian-free field. Within this framework, the time-reversal result allows us to couple two instances of the Gaussian-free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.
