Basic properties of SLE

Rohde, Steffen; Schramm, Oded

doi:10.1007/978-1-4419-9675-6_31

Cited by 283 publications

(427 citation statements)

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“…The first proof that, when κ > 8, the time reversal of an SLE κ (ρ 1 ; ρ 2 ) process is a process that belongs to the same family. It has been known for some time [29] that SLE κ itself should not have time-reversal symmetry when κ > 8. However the fact that its time reversal can be described by an SLE κ (ρ 1 ; ρ 2 ) process was not known, or even conjectured, before the current work.…”

Section: Overviewmentioning

confidence: 99%

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

Miller

Sheffield⋆

2017

Probab. Theory Relat. Fields

151

304

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We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal SLE κ (ρ 1 ; ρ 2 ) processes are time-reversible when κ < 8. Here we extend these results to wholeplane SLE κ (ρ) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ ∈ [0, 4]) to all κ ∈ [0, 8]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE κ for some κ ∈ (0, 4), and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of SLE κ where κ := 16/κ ∈ (4, ∞). By varying the boundary data we obtain, for each κ > 4, a family of space-filling variants of SLE κ (ρ) whose time reversals belong to the same family. When κ ≥ 8, ordinary SLE κ belongs to this family, and our result shows that its time-reversal is SLE κ (κ /2−4; κ /2−4). As applications of this theory, we obtain the local finiteness of CLE κ , for κ ∈ (4, 8), and describe the laws of the boundaries of SLE κ processes stopped at stopping times. Mathematics Subject Classification 60J67B Jason Miller

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Section: Overviewmentioning

confidence: 99%

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

Miller

Sheffield⋆

2017

Probab. Theory Relat. Fields

151

304

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“…It has been conjectured (see for instance [16,20,43]), based on the identification between exponents, probabilities or dimensions that one can rigorously compute for SLE processes on the one hand and the corresponding quantities that had been previously predicted using methods from theoretical physics (conformal field theory, quantum gravity or Coulomb gas methods, see for instance [6,40]) for the asymptotic behavior of the discrete models on the other hand, that the O(N ) models have a non-trivial and conformally invariant scaling limit for all N ∈ (0, 2] and that this scaling limit should be related to SLE κ curves for N = −2 cos(4π/κ), where κ ∈ (8/3, 4] if one considers the dilute O(N ) model and where κ ∈ (4,8) if one considers the dense O(N ) model. Similarly, the scaling limit of the critical FK(q)-percolation model interfaces should be non-trivial for q ∈ (0, 4] and described by SLE κ curves, for √ q = −2 cos(4π/κ), where κ ∈ [4, 8) (mind of course that cos(4π/κ) is negative for all κ ∈ (8/3, 8)).…”

Section: A Motivation From Discrete Modelsmentioning

confidence: 94%

“…More precisely, the argument was the following: If these models have a conformally invariant scaling limit, then it must be described by an SLE κ for some κ. In order to identify which value of κ is the right one, the idea in those papers was to match some computation of probabilities of events for SLE (or of critical exponents, or of dimensions) with the corresponding values that were predicted to be the correct ones for the scaling limit of the discrete model, based on theoretical physics considerations (see for instance [43] for the FK(q) conjecture based on the physics dimension predictions; another approach is related to the discrete parafermionic observable for FK models-see [53] or [13] for a detailed discussion and more references-where the spin is defined as σ = 1−(2/π ) arccos( √ q/2), and that is conjectured to correspond in the scaling limit to some SLE martingale, see for instance [61]). In the present paper, we identify the candidate value of κ using a feature that is rigorously known to hold in the discrete model (and therefore in its scaling limit, if it exists).…”

Section: Content Of the Present Papermentioning

confidence: 99%

“…10). The probability of D(ε) can be explicitly computed (it is in fact the formula that was already used by Schramm [44] in his argument mentioned at the beginning of the introduction); it is a generalization of Cardy's formula for SLE κ almost identical to that determined in [27]-see for instance [43,Lemma 6.6] or [57,Section 3]): Fig. 11.…”

Section: Case Of Non-simple Clesmentioning

confidence: 99%

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Connection Probabilities for Conformal Loop Ensembles

Miller

Werner

2018

Commun. Math. Phys.

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The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE κ (both with simple and non-simple loops, i.e., for the whole range κ ∈ (8/3, 8)), how to derive the connection probabilities in domains with four marked boundary points for a conditioned version of CLE κ which can be interpreted as a CLE κ with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a square, we prove that the probability that the two wired sides of the square hook up so that they create one single loop is equal to 1/(1 − 2 cos(4π/κ)). Comparing this with the corresponding connection probabilities for discrete O(N) models. For instance, indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is CLE κ where κ is the value in (8/3, 4] such that −2 cos(4π/κ) is equal to N (resp. the value in [4, 8) such that −2 cos(4π/κ) is equal to √ q). On the one hand, Our arguments and computations build on Dubédat's SLE commutation relations (as developed and used by Dubédat, Zhan or Bauer-Bernard-Kytölä) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field (as recently derived in works with Sheffield and with Qian). Contents

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“…It has been proven [76], [59] that with probability 1 there is a (unique) random continuous path γ (t) such that for each t ≥ 0 the domain of definition of g t is the unbounded component of H \ γ ([0, t] If D is a simply connected domain in the plane and a, b ∈ ∂D are two distinct points (or rather prime ends), then chordal SLE from a to b in D is defined as the image of γ under a conformal map from H to D taking 0 to a and ∞ to b. Though the map is not unique, the choice of the map does not effect the law of the SLE in D. This follows from the easily verified fact that up to a rescaling of time, the law of the SLE path is invariant under scaling by a positive real constant, as is the case for Brownian motion.…”

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confidence: 99%

Conformally invariant scaling limits: an overview and a collection of problems

Schramm

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.

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Basic properties of SLE

Cited by 283 publications

References 17 publications

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

Connection Probabilities for Conformal Loop Ensembles

Conformally invariant scaling limits: an overview and a collection of problems

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