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arXiv · 1504.05570

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

Bertrand Duplantier·Xuan Hieu Ho·Thanh Binh Le·Michel Zinsmeister
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Abstract

We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves in the (p,q) moment plane, which depend continuously on the SLE parameter kappa. The generalization of this integrability property to the m-fold transform of f is also given. We define a generalized integral means spectrum corresponding to the singular behavior of the mixed moments above. By inversion, it allows for 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 takes four possible forms, separated by five phase transition lines in the moment plane, whereas the average generalized spectrum of the m-fold whole-plane SLE is directly obtained from a linear map acting in that plane. We also conjecture the form of the universal generalized integral means spectrum.

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Bertrand Duplantier, Xuan Hieu Ho, Thanh Binh Le, Michel Zinsmeister. 2015-04-21. Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE. https://arxiv.org/abs/1504.05570

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