Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

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.