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Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

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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 \({{\mathbb{D}}}\) to the slit plane, the derivative moments \({\mathbb{E}(\vert f'(z) \vert^p)}\) can be written in a closed form for certain values of p depending continuously on the SLE parameter \({\kappa\in (0,\infty)}\). We generalize this property to the mixed moments, \({\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)}\), along integrability curves in the moment plane \({(p,q) \in {\mathbb{R}}^2}\) depending continuously on \({\kappa}\), 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, \({\beta(p,q;\kappa)}\), 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 \({{\mathbb{R}}^2}\). The average generalized spectrum of the m-fold whole-plane SLE is obtained directly from the m = 1 case by a linear map acting in the moment plane. We also give a conjecture for the precise form of the universal generalized integral means spectrum.

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Correspondence to Bertrand Duplantier.

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Communicated by M. Salmhofer

The first author wishes to thank the Isaac Newton Institute (INI) forMathematical 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” noPICS06769. 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.

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Duplantier, B., Ho, X.H., Le, T.B. et al. Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE. Commun. Math. Phys. 359, 823–868 (2018). https://doi.org/10.1007/s00220-017-3046-z

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