Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

TL;DR

This paper analyzes the multifractal properties of whole-plane SLE maps, deriving closed-form expressions for mixed moments and spectra, revealing phase transitions and generalizations for m-fold transforms.

Contribution

It introduces a unified framework for the generalized integral means spectrum of whole-plane SLE, including m-fold transforms and phase transition analysis.

Findings

Closed-form expressions for mixed moments along integrability curves.

Identification of four forms of the generalized spectrum with phase transitions.

Extension of the spectrum analysis to m-fold transforms and conjecture of a universal spectrum.

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.