Duality of Schramm-Loewner evolutions

Dubédat, Julien

doi:10.24033/asens.2107

Cited by 79 publications

(109 citation statements)

References 17 publications

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“…This is proved in Proposition 3 in [10], based on results in [22]. The loop measure term is identified via Proposition 2.1.…”

Section: ≥0mentioning

confidence: 99%

“…A "local to global" argument is introduced in [42] in the context of SLE reversibility. We broadly follow here the presentation in [10], one difference being in the nature of the Markov properties; there is also here additional structure (conditional independence).…”

Section: (Sle Strands) and (A Field) Thenmentioning

confidence: 99%

“…Then ( ) ≥0 is an increasing family of compact subsets of ℍ; moreover, is the unique conformal equivalence ℍ ∖ → ℍ such that (hydrodynamic normalization at ∞): [10,41]). …”

Section: Proposition 22 Under the Above Assumptionsmentioning

confidence: 99%

“…Partition functions. In this subsection we introduce partition functions of SLE (a predefinition is in [10]) and give some basic properties. These partition functions are null vectors of some canonical Virasoro representations ( [11]).…”

Section: ≥0mentioning

confidence: 99%

“…In the context of SLE reversibility, Zhan showed in [42] how to lift local couplings to global couplings. In [10], it is shown how to extend this to the framework of local commutation, in which partition functions intervene naturally. We use similar techniques here to couple in a domain one SLE strand with a free field, in conjunction with Gaussian arguments and free field properties.…”

Section: Introductionmentioning

confidence: 99%

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SLE and the free field: Partition functions and couplings

Dubédat

2009

J. Amer. Math. Soc.

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173

276

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Schramm-Loewner Evolutions (SLE) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of SLE and the free field with appropriate boundary conditions; this involves ζ \zeta -regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of SLE with the free field, showing that, in a precise sense, chordal SLE is the solution of a stochastic “differential” equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general κ > 0 \kappa >0 . This identifies SLE curves as local observables of the free field.

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“…This is proved in Proposition 3 in [10], based on results in [22]. The loop measure term is identified via Proposition 2.1.…”

Section: ≥0mentioning

confidence: 99%

Section: (Sle Strands) and (A Field) Thenmentioning

confidence: 99%

“…Then ( ) ≥0 is an increasing family of compact subsets of ℍ; moreover, is the unique conformal equivalence ℍ ∖ → ℍ such that (hydrodynamic normalization at ∞): [10,41]). …”

Section: Proposition 22 Under the Above Assumptionsmentioning

confidence: 99%

Section: ≥0mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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SLE and the free field: Partition functions and couplings

Dubédat

2009

J. Amer. Math. Soc.

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173

276

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Integrability of SLE via conformal welding of random surfaces

Ang,

Holden,

Sun

2023

Comm Pure Appl Math

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We demonstrate how to obtain integrability results for the Schramm‐Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating‐of‐trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called . Our proof is built on two connections between SLE, LCFT, and mating‐of‐trees. Firstly, LCFT and mating‐of‐trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper Ang, Holden and Sun (2023). Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating‐of‐trees based on these two connections.

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Oded Schramm: From Circle Packing to SLE

Rohde

2011

Selected Works of Oded Schramm

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Duality of Schramm-Loewner evolutions

Cited by 79 publications

References 17 publications

SLE and the free field: Partition functions and couplings

SLE and the free field: Partition functions and couplings

Integrability of SLE via conformal welding of random surfaces

Oded Schramm: From Circle Packing to SLE

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