The SLE loop via conformal welding of quantum disks

Ang, Morris; Holden, Nina; Sun, Xin

doi:10.1214/23-ejp914

Cited by 5 publications

(31 citation statements)

References 38 publications

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“…As explained at the end of [CCM20, Section 1], Theorem 3.10 follows by combining results in [MSW22, CCM20, BBCK18]. Further details are given in Appendix A.2 of the first arXiv version of [AS21].…”

Section: Proof Of Theorem 12mentioning

confidence: 91%

“…The analog of Theorem 1.1 for γ ∈ ( 8/3, 2) and κ = γ 2 ∈ (8/3, 4) is immediate from [MSW22, Theorem 1.1] (see Theorem 3.1 below). We will deduce Theorem 1.1 from this statement, plus a minor extension coming from [AS21], by taking the limit as γ → 2 − . This is similar to the strategy used to establish relationships between critical LQG and SLE 4 in [HP21,AHPS21].…”

Section: Introductionmentioning

confidence: 88%

“…An alternative definition of the γ-LQG disk for γ ∈ (0, 2), based on conformal field theory, is given in [HRV18]. The definition there and the one in Definition 2.3 are proven to be equivalent in [Cer21] (see also [AHS21]).…”

Section: The Lqg Diskmentioning

confidence: 99%

“…In the subcritical case c L > 25, the LQG disk is defined similarly except that we take B to be a Brownian motion with negative drift conditioned to stay negative, instead of a 3d Bessel process, and we weight the law of Φ 0 by ν Φ 0 (∂S) 4/γ 2 −1 , see [DMS21, Section 4.5]. There is also an alternative definition of the LQG disk given in [HRV18], which is proven to be equivalent to the one in [DMS21] in [Cer21] (see also [AHS21] for an alternative proof).…”

Section: The Supercritical Lqg Diskmentioning

confidence: 99%

“…The relationships between SLE and LQG in the subcritical and critical phases have a huge number of applications, to such topics as SLE and of LQG individually, conformal field theory (see, e.g., [AS21,ARSZ23]), the geometry of random planar maps (see, e.g., [GHS20]), random permutations (see, e.g., [Bor21]), and the moduli of random surfaces [ARS22]. Many of these applications are surveyed in [GHS19].…”

mentioning

confidence: 99%

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Liouville quantum gravity surfaces with boundary as matings of trees

Ang

Gwynne

2021

Ann. Inst. H. Poincaré Probab. Statist.

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Consider a critical (γ = 2) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter κ = 4. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE 4 loops are conditionally independent critical LQG disks given the LQG lengths of the loops. We also show that the joint law of the LQG lengths of the loops is described in terms of the jumps of a certain 3/2-stable process. Our proofs are via a limiting argument based on the analogous statements for γ ∈ ( 8/3, 2) and κ = γ 2 ∈ (8/3, 4) which were proven by Miller, Sheffield, and Werner (2020). Our results are used in the construction of a coupling of supercritical LQG with CLE 4 in another paper by the same authors.

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Section: Proof Of Theorem 12mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 88%

Section: The Lqg Diskmentioning

confidence: 99%

Section: The Supercritical Lqg Diskmentioning

confidence: 99%

mentioning

confidence: 99%

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Liouville quantum gravity surfaces with boundary as matings of trees

Ang

Gwynne

2021

Ann. Inst. H. Poincaré Probab. Statist.

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Conformal covariance of connection probabilities and fields in 2D critical percolation

Camia

2023

Comm Pure Appl Math

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Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well‐defined scaling limit for every . Moreover, the limiting functions transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and for any . This implies that and , for some constants C2 and C3.We also define a site‐diluted spin model whose n‐point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well‐defined scaling limit with the same properties as the functions . In particular, . We prove that the magnetization field associated with this spin model has a well‐defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four‐point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four‐arm events and can be identified with the so‐called “four‐leg operator” of conformal field theory.

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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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The SLE loop via conformal welding of quantum disks

Cited by 5 publications

References 38 publications

Liouville quantum gravity surfaces with boundary as matings of trees

Liouville quantum gravity surfaces with boundary as matings of trees

Conformal covariance of connection probabilities and fields in 2D critical percolation

Integrability of SLE via conformal welding of random surfaces

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