The Coefficient Problem and Multifractality of Whole-Plane SLE and LLE

TL;DR

This paper investigates the multifractal properties and coefficient expectations of whole-plane SLE and LLE processes, revealing phase transitions and connections to quantum gravity and radial SLE exponents.

Contribution

It provides exact results for coefficient expectations, variances, and derivative moments of whole-plane SLE and LLE, and explores their multifractal spectra and phase transitions.

Findings

Exact expectations and variances for coefficients of whole-plane SLE and LLE.

Identification of a phase transition in the integral means spectrum.

Connection between multifractal spectra and quantum gravity exponents.

Abstract

We revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, L\'evy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the m-fold conformal maps of whole-plane SLEs or L\'evy-Loewner Evolutions (LLEs). We also study the (averaged) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a certain moment order, at which one goes from the bulk SLE expected integral means spectrum, as established by Beliaev and Smirnov, to a new integral means spectrum. The latter is furthermore shown to be intimately related, via the associated packing spectrum, to radial SLE derivative exponents, and to local SLE tip multifractal exponents obtained from quantum gravity. This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map.