Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

Duplantier, Bertrand; Binder, Ilia

doi:10.1016/j.nuclphysb.2008.05.020

Cited by 14 publications

(19 citation statements)

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“…It sets up the martingale techniques needed for dealing with mixed moments. Section 3 uses them for the study of the complex one-point function (a) in (12), which is shown to obey a simple differential equation in complex variable z. This leads to Theorem 3.1, which establishes a closed form for this function along the integrability parabola R in the (p, q)-plane.…”

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confidence: 98%

“…This leads to Theorem 3.1, which establishes a closed form for this function along the integrability parabola R in the (p, q)-plane. Section 4 is concerned with the moduli one-point function (b) in (12), and more generally, with the SLE two-point function G(z 1 ,z 2 ). A PDE in (z 1 ,z 2 ) is derived for G(z 1 ,z 2 ), which yields a proof of Theorem 4.2 establishing closed form expressions for G for all (p, q) ∈ R. Section 4.4 deals with the generalization of the previous integrability results to the m-fold symmetric transforms f [m] , m ∈ Z \ {0}, of the whole-plane SLE map f .…”

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confidence: 99%

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Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

Duplantier

et al. 2017

Commun. Math. Phys.

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Abstract. It has been shown that for f an instance of the whole-plane SLE unbounded conformal map from the unit disk D to the slit plane, the derivative moments E(|f (z)| p ) can be written in a closed form for certain values of p depending continuously on the SLE parameter κ ∈ (0, ∞). We generalize this property to the mixed moments, E |f (z)| p |f (z)| q , along integrability curves in the moment plane (p, q) ∈ R 2 depending continuously on κ, by extending the so-called Beliaev-Smirnov equation to this case. The generalization of this integrability property to the m-fold transform of f is also given. We define a novel generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the above mixed moments. By inversion, it allows a unified description of the unbounded interior and bounded exterior versions of whole-plane SLE, and of their m-fold generalizations. The average generalized spectrum of whole-plane SLE is found to take four possible forms, separated by five phase transition lines in the moment plane R 2 . The average generalized spectrum of the m-fold whole-plane SLE is directly obtained from the m = 1 case by a linear map acting in the moment plane. We also conjecture the precise form of the universal generalized integral means spectrum.Date: April 24, 2017. Key words and phrases. Whole-plane SLE, logarithmic coefficients, Beliaev-Smirnov equation, generalized integral means spectrum, universal spectrum.The first author wishes to thank the Isaac Newton Institute (INI) for Mathematical Sciences at Cambridge University, where part of this work was completed, for its hospitality and support during the 2015 program "Random Geometry", supported by EPSRC Grant Number EP/K032208/1. B.D. also gratefully acknowledges the support of a Simons Foundation fellowship at INI during the Random Geometry program. B.D. acknowledges financial support from the French Agence Nationale de la Recherche via the grant ANR-14-CE25-0014 "GRAAL"; he is also partially funded by the CNRS Projet international de coopération scientifique (PICS) "Conformal Liouville Quantum Gravity" n o PICS06769. The research by the second author is supported by a joint scholarship from MENESR and Région Centre; the third author is supported by a scholarship of the Government of Vietnam. B.D. and M.Z. are partially funded by the CNRS-insmi Groupement de Recherche (GDR 3475) "Analyse Multifractale". BD and MZ also gratefully acknowledge the continuous hospitality of the Institut Henri Poincaré in Paris. Figure 1. Loewner map z → f t (z) from D to the slit domain Ω t = C\γ([t, ∞)) (here slit by a single curve γ([t, ∞)) for SLE κ≤4 ). One has f t (0) = 0, ∀t ≥ 0. At t = 0, the driving function λ(0) = 1, so that the image of z = 1 is at the tip γ(0) = f 0 (1) of the curve. [From Ref. [14]].

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confidence: 98%

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Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

Duplantier

et al. 2017

Commun. Math. Phys.

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“…On the other hand this bounded number of Whitney squares can be in turn covered by a uniformly bounded number of connected Whitney squares in D. Thus also the ratios of distances of f (z), f (w), f (u), f (v) from the boundary are bounded by constants. It follows again from the distortion theorems (14) that also the ratios of the different f (·) are bounded, giving us the claim.…”

Section: An Estimate On the Green's Functionmentioning

confidence: 65%

“…Very near the curve, this distortion is given by unwinding the SLE curve back to zero. One can also think that this definition of winding gives the amount that a curve from the infinity needs to wind to access the point z. Asymptotically near the curve, this version of winding should coincide with the geometric winding up to some bounded constants [14].…”

Section: Introduction and Resultsmentioning

confidence: 99%

KPZ relation does not hold for the level lines and $$\hbox {SLE}_\kappa $$ flow lines of the Gaussian free field

Aru

2014

Probab. Theory Relat. Fields

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In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation in a natural way, shedding new light on all of them. Our principal result shows that the level lines and the SLE κ flow lines of the Gaussian free field do not satisfy the usual KPZ relation. In order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.

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“…There are a number of other quantities related to the multifractal spectrum which have been computed either rigorously or non-rigorously. These include the winding spectrum (predicted in [DB02,DB08]), higher multifractal spectra depending on the derivative behavior on both sides of the curve (predicted in [Dup03]), the integral means spectrum (rigorous computations of different versions given in [BS09a, GMS17, DNNZ12, LY13, LY14]), the multifractal spectrum at the tip (computed in [JVL12]), and the boundary multifractal spectrum (computed in [ABJ15]).…”

Section: Overviewmentioning

confidence: 99%

Dimension transformation formula for conformal maps into the complement of an SLE curve

Gwynne

Holden

Miller

2019

Probab. Theory Relat. Fields

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We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of R and the Hausdorff dimension of its image under a conformal map from the upper half plane to a complementary connected component of an SLEκ curve for κ = 4. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula of Gwynne-Holden-Miller (2015). As an intermediate step we obtain a KPZ formula which relates the Euclidean dimension of a subset of an SLEκ curve for κ ∈ (0, 4) to the dimension of the same set with respect to the γ-quantum length measure on the curve induced by an independent Gaussian free field, γ = √ κ. As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an SLEκ curve for κ ∈ (0, 4) ∪ (4, 8) and the dimension of the same set with respect to the γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, γ = √ κ ∧ (4/ √ κ). arXiv:1603.05161v2 [math.PR] 27 Oct 2017 Φ κ (d) := 1 32κ 4 + κ − (4 + κ) 2 − 16κd 12 + 3κ + (4 + κ) 2 − 16κd . (1.2)

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Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

Cited by 14 publications

References 58 publications

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

KPZ relation does not hold for the level lines and $$\hbox {SLE}_\kappa $$ flow lines of the Gaussian free field

Dimension transformation formula for conformal maps into the complement of an SLE curve

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