Logarithmic Coefficients and Multifractality of Whole-Plane SLE

Bertrand Duplantier (1) , Xuan Hieu Ho (2) , Thanh Binh Le (3) , Michel Zinsmeister (4, 5)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
3 E3S - Supélec Sciences des Systèmes
4 LPMC - Laboratoire de physique de la matière condensée
5 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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. 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.

Domains: Mathematical Physics [math-ph]

Bertrand Duplantier, Xuan Hieu Ho, Thanh Binh Le, Michel Zinsmeister. Logarithmic Coefficients and Multifractality of Whole-Plane SLE. 2016. ⟨cea-01251725⟩