Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

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

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 UO - Université d'Orléans

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.

Domaines: Physique [physics]

Bertrand Duplantier, Xuan Hieu Ho, Thanh Binh Le, Michel Zinsmeister. Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE. Communications in Mathematical Physics, 2018, 359 (3), pp.823-868. ⟨10.1007/s00220-017-3046-z⟩. ⟨cea-04409263⟩