Extreme nesting in the conformal loop ensemble

Miller, Jason; Watson, Samuel S.; Wilson, David B.

doi:10.1214/14-aop995

Cited by 22 publications

(31 citation statements)

References 19 publications

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“…is dominated by twice a geometric random variable, and by Lemma 2.8 in [10] together with the Koebe quarter theorem we have J ∩ z,ε − J ∩ z,r is order log(r/ε) except with probability O((ε/r ) c 1 ), for some constant c 1 > 0 (depending on κ). Therefore, except with probability…”

Section: Lemma 42 For Any κ ∈ (8/3 8) and J ∈ N There Is A Positivmentioning

confidence: 80%

“…We show that z → S z (ε) − E[S z (ε)] converges as ε → 0 to a distribution we call the weighted nesting field. When κ = 4 and μ is a signed Bernoulli distribution, the weighted nesting field is the GFF [9,10]. Our result serves to generalize this construction to other values of κ ∈ (8/3, 8) and weight measures μ.…”

Section: Introductionmentioning

confidence: 85%

“…We refer the reader to the preliminaries section [10] for an introduction to CLE. We begin by reminding the reader of the Koebe distortion theorem and the Koebe quarter theorem.…”

Section: Basic Cle Estimatesmentioning

confidence: 99%

“…By [10,Lemma 3.6], except with probability exponentially small in x, both 1 and 2 contain a loop surrounding B(0, 1) which is contained in B(0, e x/4 ). It is possible that ψ maps a loop of 1 surrounding D to a loop of 2 intersecting D or vice versa.…”

Section: Full-plane Cle κ Is Invariant Under Scalings Translations mentioning

confidence: 99%

“…When κ = 4, σ = √ π/2, and μ = μ B where μ B ({σ }) = μ B ({−σ }) = 1/2 (as in Theorem 1.2 of [10]) the distribution h of Theorem 1.1 is that of a GFF on D [9]. The existence of the distributional limit for other values of κ was posed in [12,Problem 8.2].…”

Section: Introductionmentioning

confidence: 99%

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The conformal loop ensemble nesting field

Miller

Watson

Wilson

2015

Probab. Theory Relat. Fields

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The conformal loop ensemble CLE κ with parameter 8/3 < κ < 8 is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. We show that the number of loops surrounding an ε-ball (a random function of z and ε) minus its expectation converges almost surely as ε → 0 to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points.

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Section: Lemma 42 For Any κ ∈ (8/3 8) and J ∈ N There Is A Positivmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 85%

“…We refer the reader to the preliminaries section [10] for an introduction to CLE. We begin by reminding the reader of the Koebe distortion theorem and the Koebe quarter theorem.…”

Section: Basic Cle Estimatesmentioning

confidence: 99%

Section: Full-plane Cle κ Is Invariant Under Scalings Translations mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The conformal loop ensemble nesting field

Miller

Watson

Wilson

2015

Probab. Theory Relat. Fields

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Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

Camia

Stein

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Dimension of the SLE Light Cone, the SLE Fan, and $${{\rm SLE}_\kappa(\rho)}$$ SLE κ ( ρ ) for $${\kappa \in (0,4)}$$ κ ∈ ( 0 , 4 ) and $${\rho \in}$$ ρ ∈ $${\big[{\tfrac{\kappa}{2}}-4,-2\big)}$$ [ κ 2 - 4 , - 2 )

Miller

2018

Commun. Math. Phys.

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Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix κ ∈ (0, 4). The SLE κ light cone L(θ) of h with opening angle θ ∈ [0, π] is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field e ih/χ , χ = 2 √ κ − √ κ 2 , with angles in [− θ 2 , θ 2 ]. We derive the Hausdorff dimension of L(θ). If θ = 0 then L(θ) is an ordinary SLE κ curve (with κ < 4); if θ = π then L(θ) is the range of an SLE κ curve (κ = 16/κ > 4). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE. We also consider SLE κ (ρ) processes, which were originally only defined for ρ > − 2, but which can also be defined for ρ ≤ −2 using Lévy compensation. The range of an SLE κ (ρ) is qualitatively different when ρ ≤ −2. In particular, these curves are selfintersecting for κ < 4 and double points are dense, while ordinary SLE κ is simple. It was previously shown (Miller and Sheffield in Gaussian free field light cones and SLE κ (ρ), 2016) that certain SLE κ (ρ) curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLE κ (ρ) for all values of ρ. Finally, we show that the Hausdorff dimension of the so-called SLE κ fan is the same as that of ordinary SLE κ .

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Extreme nesting in the conformal loop ensemble

Cited by 22 publications

References 19 publications

The conformal loop ensemble nesting field

The conformal loop ensemble nesting field

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

Dimension of the SLE Light Cone, the SLE Fan, and $${{\rm SLE}_\kappa(\rho)}$$ SLE κ ( ρ ) for $${\kappa \in (0,4)}$$ κ ∈ ( 0 , 4 ) and $${\rho \in}$$ ρ ∈ $${\big[{\tfrac{\kappa}{2}}-4,-2\big)}$$ [ κ 2 - 4 , - 2 )

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