Integral means spectrum of whole-plane SLE

TL;DR

This paper analyzes the integral means spectrum of whole-plane SLE, revealing phase transitions related to the SLE origin and tip, advancing understanding of harmonic measure on SLE curves.

Contribution

It provides a detailed mathematical analysis of the integral means spectrum for bounded whole-plane SLE, identifying phase transitions at different moments.

Findings

Phase transition at the SLE origin for low moments.

Transition to the SLE tip spectrum occurs earlier than the origin transition.

Complete characterization of the harmonic measure's fine structure on SLE curves.

Abstract

We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated by Beliaev and Smirnov, as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and Loutsenko and Yermolayeva, a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE studied here, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.