Conformal Restriction, Highest-Weight Representations and SLE

Friedrich, R.; Werner, Wendelin

doi:10.1007/s00220-003-0956-8

Cited by 78 publications

(125 citation statements)

References 45 publications

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“…In fact, a similar result holds for all κ ∈ (0, 8/3) : Adding Brownian loops to an SLE κ gives a sample of a conformal restriction measure, as defined in [14]. This fact can be related to some representations of infinite-dimensional Lie Algebras [8]. In a way, this shows that SLE κ for κ ∈ (0, 8/3] could also be characterized implicitly and globally via planar Brownian motions : It is the only simple curve such that if one adds a certain density of Brownian loops, one gets the same hull as the union of some Brownian motions (all these aspects relating restriction measures to SLE are reviewed in [25]).…”

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confidence: 83%

SLEs as boundaries of clusters of Brownian loops

Werner

2003

Comptes Rendus. Mathématique

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RésuméIn this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLEκ-type curves where κ ∈ (8/3, 4] and c related by the usual relation c = (3κ − 8)(6 − κ)/2κ (i.e. c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLEκ loops, that behaves "nicely" under perturbation of the surface and is related to various aspects of conformal field theory and representation theory. BackgroundThe goal of this paper is announce some results that relate the Brownian loop soup introduced in [16] to the Schramm-Loewner Evolutions (SLE) (in particular for values of the parameter between 8/3 and 4) and random families of disjoint SLE loops (as they might appear in the scaling limit of many 2d statistical physics models, and in conformal field theory). A more complete paper [26] on the same subject (with proofs, that discusses also various consequences of this approach) is in preparation.SLE processes have been introduced by Schramm in [21], building on the observation that they are the only processes that have a certain (conformal) Markovian-type property. There is a one-dimensional family of SLEs (indexed by a positive real parameter κ), and they are the only possible candidates for the scaling limits of interfaces for two-dimensional critical systems that are believed to be conformally invariant. This definition of SLE via Loewner's equation is a dynamic one-dimensional construction : One basically describes the law of η[t, t+ dt] given η [0, t], and integrates this with respect to t. See e.g. [21,20,10,23] for an introduction to (chordal) SLE. Furthermore, it is worthwhile stressing that this construction describes one interface, corresponding to specific boundary conditions in the discrete model, but that it does in general not give immediately access to the "complete scaling limit" of the system. This raises the following question : Is there a simple and natural way to define at once a whole family 1 of SLE loops in a domain that might describe simultaneously all boundaries of clusters ?In [14], a different characterization of the SLE 8/3 random curve was derived. It is shown to be the unique random curve in a domain, that satisfies a certain conformal restriction property. This characterization is "global" and does not use (directly) the Markovian property. It also enabled to identify this curve with the outer boundary of a certain reflected Brownian motion [14], and with the outer boundary of a certain union of Brownian excursions [25]. Hence, it is geometrically possible to construct SLE 8/3 from planar Brownian motions (recall also that SLE 8/3 is conjectured to be the scaling limit of the half-plane self-avoiding walk [13]).When κ = 2, SLE has been proved in [12] to be the scaling limit ...

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confidence: 83%

SLEs as boundaries of clusters of Brownian loops

Werner

2003

Comptes Rendus. Mathématique

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“…It also turned out (see e.g. [13,3]) that this chordal restriction property of SLE with parameter 8/3 had interpretations in terms of boundary conformal field theory and that it yields a simple definition of the SLE 8/3 in non-simply connected domains. Also, a by-product of the results of [21] was the fact that the Brownian outer boundary looked "locally" like that of a critical percolation boundary, and also like an SLE 8/3 , because of global identities in law between outer boundaries of the union of five Brownian excursions and that of the union of eight SLE 8/3 's.…”

Section: Further Motivation and Backgroundmentioning

confidence: 96%

The conformally invariant measure on self-avoiding loops

Werner

2007

J. Amer. Math. Soc.

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We show that there exists a unique (up to multiplication by constants) and natural measure on simple loops in the plane and on each Riemann surface, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surface S ′ S’ that is contained in another Riemann surface S S is just the measure on S S restricted to those loops that stay in S ′ S’ ). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.

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“…Then, either the path γ hits it also (and this probability is given by B N +1 ) or it avoids it (and the probability that it hits the N other ones is now given in terms of B N and the conformal mapping from H \ [x, x + iε] onto H. This relation can be written as: for some operators L N . This equation can then be rephrased in terms of a highest-weight representation of A with highest-weight α, see [21] for details. Basically, one shows that these L n 's, when defined on appropriate functions, do satisfy the same commutation relation as the l n .…”

Section: Relation With Restrictionmentioning

confidence: 99%

Conformal restriction and related questions

Werner¹

2005

Probab. Surveys

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These notes are based on mini-courses given in July 2003 at the University of St-Andrews, and at the ICMS in Edinburgh. The goal of these lectures is to give a self-contained sketchy and heuristic survey of the recent results concerning conformal restriction, that were initiated in our joint work with Greg Lawler and Oded Schramm [37,36], and further investigated in the papers [21,57,41,13]. It also finds some of its roots in the earlier "pre-SLE" paper [40]. If the reader wants full proofs and cleaner statements, (s)he is advised to consult these original papers. We will also not go into the details of the construction of the Schramm-Loewner Evolution (SLE), referring the motivated and interested reader to [56,29] and the references therein for more details. These notes can be viewed as complementary to my Saint-Flour notes [56] (one can read either of them before the other), and are maybe aimed at a less probabilistic audience.The papers dealing with the restriction property all take the existence and properties of SLE for granted, and are therefore maybe difficult to read if one is not already acquainted to the SLE background. We will try in these lectures to conversely use the intuition and ideas building on the restriction properties and self-avoiding walks to shed some light onto some of the properties of SLE, and its relation to some ideas from conformal field theory. We hope that this can be useful in the theoretical physics community as well.Almost all results in these notes are borrowed from the existing abovementioned papers, but there are a few new ones, such as for instance the construction of the one-sided restriction measures via Poissonian clouds of Brownian excursions.

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Conformal Restriction, Highest-Weight Representations and SLE

Cited by 78 publications

References 45 publications

SLEs as boundaries of clusters of Brownian loops

SLEs as boundaries of clusters of Brownian loops

The conformally invariant measure on self-avoiding loops

Conformal restriction and related questions

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