Non-simple SLE curves are not determined by their range

Miller, Jason; Sheffield⋆, Scott; Werner, Wendelin

doi:10.4171/jems/930

Cited by 26 publications

(32 citation statements)

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“…Note that for general β, one can choose the status of a CLE κ hole only once the percolation interfaces does indeed hit it, which explains the relation to the so-called side-swapping SLE κ (κ − 6) processes, that will be at the heart of our analysis. In the subsequent paper [40], we shall prove that (when κ ∈ (8/3, 4)), these percolation interfaces are indeed still random (when one conditions on the CLE κ ) for all choices of β ∈ [−1, 1]. This means that this percolation process captures additional randomness that is located "inside" the CLE κ carpet.…”

Section: 22mentioning

confidence: 77%

“…In the subsequent paper [40], we shall prove that when κ ∈ (8/3, 4), the CPI are not deterministically determined from the labeled CLE, meaning that it captures additional randomness that is not present in the labeled CLE. This contrasts with the case κ = 4 that we will discuss in the next section.…”

Section: Conformal Percolation In Cle β κ Carpets For κ <mentioning

confidence: 93%

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Cle Percolations

Miller

Sheffield⋆

Werner

2017

Forum of Mathematics, Pi

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173

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Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set -a random and conformally invariant analog of the Sierpinski carpet or gasket.In the present paper, we derive a direct relationship between the CLEs with simple loops (CLE κ for κ ∈ (8/3, 4), whose loops are Schramm's SLE κ -type curves) and the corresponding CLEs with nonsimple loops (CLE κ with κ := 16/κ ∈ (4, 6), whose loops are SLE κ -type curves). This correspondence is the continuum analog of the Edwards-Sokal coupling between the q-state Potts model and the associated FK random cluster model, and its generalization to noninteger q.Like its discrete analog, our continuum correspondence has two directions. First, we show that for each κ ∈ (8/3, 4), one can construct a variant of CLE κ as follows: start with an instance of CLE κ , then use a biased coin to independently color each CLE κ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE κ loops as interfaces of a continuum analog of critical Bernoulli percolation within CLE κ carpets -this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by SLE 6 and CLE 6 .These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized SLE κ (ρ) curves for ρ < −2, such as their decomposition into collections of SLE κ -type 'loops' hanging off of SLE κ -type 'trunks', and vice versa (exchanging κ and κ ). We also define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of

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Section: 22mentioning

confidence: 77%

Section: Conformal Percolation In Cle β κ Carpets For κ <mentioning

confidence: 93%

Cle Percolations

Miller

Sheffield⋆

Werner

2017

Forum of Mathematics, Pi

Self Cite

173

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“…In the special critical case, this CLE (which is CLE 4 ) can also be constructed as level lines of the two-dimensional Gaussian free field (GFF), see Miller-Sheffield [10], or [1]. There exists also a coupling of the GFF with the other CLEs, but it is much more involved and somewhat less natural (it for instance relies on various choices, such as the choice of a root on the boundary of the domain), see for instance the comments in [11,12]. This suggests that understanding better the coupling of CLEs with subcritical loop-soups is actually of interest in order to understand better CLEs themselves.…”

Section: General Goalmentioning

confidence: 99%

Conditioning a Brownian loop-soup cluster on a portion of its boundary

Qian¹

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We show that if one conditions a cluster in a Brownian loop-soup L (of any intensity) in a two-dimensional domain by a portion ∂ of its outer boundary, then in the remaining domain, the union of all the loops of L that touch ∂ satisfies the conformal restriction property while the other loops in L form an independent loop-soup. This result holds when one discovers ∂ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch ∂.Our results can be viewed as an extension of some of the results in our paper [14] in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained in [14] that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds. RésuméDans le présent article, nous étudions certaines propriétés des amas de lacets browniens, dans une soupe de lacets browniens d'intensité c dans un domaine du plan, pour toute intensité c ≤ 1.Notre principal résultat dit que si l'on découvre de manière Markovienne une portion ∂ du bord extérieur d'un tel amas, alors dans le domaine restant, la loi conditionnelle de l'union de tous les lacets dans L qui touchent ∂ satisfait la propriété de restriction conforme tandis que les autres lacets dans L forment une soupe de lacets indépendante. Ceci implique en particulier l'existence d'une transition de phase à c = 14/15 pour la connectivité de l'ensemble des lacets qui touchent ∂.Nos résultats constituent une extension de certaines résultats de notre papier [14] dans les deux directions suivantes : Dans [14], un cluster de lacets est conditionné par son bord extérieur entier tandis que nous découvrons ici seulement une partie de ce bord. En outre, dans [14], nous expliquons que la description que nous donnons de l'ensemble des lacets qui touchent ce bord via un processus ponctuel de Poisson d'excursions est spécifique au cas de la soupe de lacets critique (c = 1), nous montrons ici que dans les cas sous-critiques c < 1, une propriété plus faible de restriction conforme reste néanmoins vraie. MSC: 60J65, 60J67, 60K35• The loops in Γ b alone almost surely do not hook up into one single cluster when c ∈ (14/15, 1].However, they are almost surely hooked up into one single cluster by the loops that are of positive distance from the outer-boundary of that outermost cluster.The Brownian loop-soups. The Brownian loop-soups have been introduced by Lawler and Werner in [8]. A Brownian loop-soup Γ D in a domain D is a Poisson point process of unrooted Brownian loops of intensity c, restricted to the lo...

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“…The existence of 2-SLE κ was proved in [3] for κ ∈ (0, 4] using Brownian loop measure and in [11,9] for κ ∈ (4, 8) using flow line theory. The uniqueness of 2-SLE κ (for a fixed domain and link pattern) was proved in [10] (for κ ∈ (0, 4]) and [12] (for κ ∈ (4, 8)) using an ergodicity argument.…”

Section: Chordal Slementioning

confidence: 99%

Two-Curve Green’s Function for 2-SLE: The Interior Case

Zhan

2020

Commun. Math. Phys.

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We prove that for a 2-SLE κ pair (η 1 , η 2 ) in a simply connected domain D, whose boundary is C 1 near z 0 ∈ ∂D, there is some α > 0 such that lim r→0 + r −α P[dist(z 0 , η j ) < r, j = 1, 2] converges to a positive number, called the boundary two-curve Green's function. The exponent α equals 2( 12 κ − 1) if z 0 is not one of the endpoints of η 1 and η 2 ; and otherwise equals 12 κ − 1. We also prove the existence of the boundary (one-curve) Green's function for a single-boundary-force-point SLE κ (ρ) curve, for κ and ρ in some range. In addition, we find the convergence rate and the exact formula of the above Green's functions up to multiplicative constants. To derive these results, we construct a family of two-dimensional diffusion processes, and use orthogonal polynomials to obtain their transition density.converges to a positive number, where the exponent α equals α 0 := (12−κ)(κ+4) 8κ . The limit G(z 0 ) is called the (interior) two-curve Green's function for (η 1 , η 2 ). The paper [22] also derived the convergence rate and the exact formula of G(z 0 ) up to an unknown constant.In this paper we study the limit in the case that z 0 ∈ ∂D, assuming that ∂D is C 1 near z 0 , for some suitable exponent α. Below is our first main theorem.Theorem 1.1. Let κ ∈ (0, 8). Let (η 1 , η 2 ) be a 2-SLE κ in a simply connected domain D. Let z 0 ∈ ∂D. Suppose ∂D is C 1 near z 0 . We have the following results. PreliminaryWe first fix some notation. Let H = {z ∈ C : Im z > 0}. For z 0 ∈ C and S ⊂ C, let rad z 0 (S) = sup{|z − z 0 | : z ∈ S ∪ {z 0 }}. If a function f is absolutely continuous on I, and f ′ = g a.e. on I, then we write f ′ ae = g on I. This means that f (x 2 ) − f (x 1 ) = Let K be a non-empty H-hull. Let K doub = K ∪ {z : z ∈ K}, where K is the closure of K, and z is the complex conjugate of z. By Schwarz reflection principle, there is a compact set S K ⊂ R such that g K extends to a conformal map from C \ K doub onto C \ S K . LetExample. Let x 0 ∈ R, r > 0. Then H := {z ∈ H : |z − x 0 | ≤ r} is an H-hull with g H (z) = z + r 2 z−x 0 , hcap(H) = r 2 , a H = x 0 − r, b H = x 0 + r, H doub = {z ∈ C : |z − x 0 | ≤ r}, c H = x 0 − 2r, d H = x 0 + 2r. The next proposition combines Lemmas 5.2 and 5.3 of [28].

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Non-simple SLE curves are not determined by their range

Cited by 26 publications

References 30 publications

Cle Percolations

Cle Percolations

Conditioning a Brownian loop-soup cluster on a portion of its boundary

Two-Curve Green’s Function for 2-SLE: The Interior Case

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