SLE and the free field: Partition functions and couplings

Dubédat, Julien

doi:10.1090/s0894-0347-09-00636-5

Cited by 170 publications

(283 citation statements)

References 44 publications

(136 reference statements)

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“…3.3. (As stated in [6], the theorem was conditional on the existence of solutions to a certain SDE, but we will prove this existence in Sect. 2.)…”

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 98%

“…2,3,4,5). Several works in recent years have addressed special cases and variants of this question [6,8,10,19,28,31,37] and have shown that in certain circumstances there is a sense in which the paths are well-defined (and uniquely determined) by h, and are variants of the Schramm-Loewner evolution (SLE). In this article, we will focus on the case that z is point on the boundary of the domain where h is defined and establish a more general set of results.…”

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 99%

“…We give a complete description of the conditional law of h given a finite collection of (possibly intersecting) flow lines. (The conditional law of h given multiple flow line segments is discussed in [6], but the results there only apply to non-intersecting segments. Extending these results requires, among other things, ruling out pathological behavior of the conditional expectation of the field-given the paths-near points where the paths intersect.)…”

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 99%

“…One can interpret the e ih direction 6 The set of flow lines in D will be the pullback via a conformal map ψ of the set of flow lines in D provided h is transformed to a new function h in the manner shown as "north" and the e i(h+π/2) direction as "west", etc. Then h determines a way of assigning a set of compass directions to every point in the domain, and a ray is determined by an initial point and a direction.…”

Section: Background and Settingmentioning

confidence: 99%

“…That is, for each t, g t is the unique conformal transformation of the unbounded connected component of H\η([0, t]) to H that looks like the identity at infinity: lim z→∞ |g t (z) − z| = 0. Loewner's theorem says that g t is a solution to the equation 6) where W t = g t (η(t)), provided η is parameterized appropriately. It will be convenient for us to consider the centered Loewner flow f t = g t − W t of η in place of g t .…”

Section: Background and Settingmentioning

confidence: 99%

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Imaginary geometry I: interacting SLEs

Miller

Sheffield⋆

2016

Probab. Theory Relat. Fields

208

483

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Fix constants χ > 0 and θ ∈ [0, 2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e i(h/χ +θ) starting at a fixed boundary point of the domain. Letting θ vary, one obtains a family of curves that look locally like SLE κ processes with κ ∈ (0, 4) (where2 ), which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE 16/κ ) within the same geometry using ordered "light cones" of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE κ (ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE 16/κ (ρ) processes. Mathematics Subject Classification 60J67B Jason Miller

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“…3.3. (As stated in [6], the theorem was conditional on the existence of solutions to a certain SDE, but we will prove this existence in Sect. 2.)…”

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 98%

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 99%

Section: η (T) = E I(h(η(t))/χ +θ)mentioning

confidence: 99%

Section: Background and Settingmentioning

confidence: 99%

Section: Background and Settingmentioning

confidence: 99%

See 3 more Smart Citations

Imaginary geometry I: interacting SLEs

Miller

Sheffield⋆

2016

Probab. Theory Relat. Fields

208

483

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General β‐Jacobi Corners Process and the Gaussian Free Field

Borodin

Gorin

2014

Comm Pure Appl Math

118

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We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general β Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the β → ∞ limit.

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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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SLE and the free field: Partition functions and couplings

Cited by 170 publications

References 44 publications

Imaginary geometry I: interacting SLEs

Imaginary geometry I: interacting SLEs

General β‐Jacobi Corners Process and the Gaussian Free Field

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

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