A conformally invariant growth process of SLE excursions

TL;DR

This paper introduces a conformally invariant growth process of SLE excursions in the unit disk, establishing its existence for certain parameters, analyzing its fractal properties, and constructing related conformal fields.

Contribution

It constructs the conformal growth process of SLE excursions, proves its existence for ppa in [0,4), and analyzes its fractal and conformal invariance properties.

Findings

CGE(ppa) exists iff ppa in [0,4)

Hausdorff dimension of attached arcs is 1+ppa/8

Constructs conformally invariant random fields based on CGE(ppa)

Abstract

We construct an aggregation process of chordal SLE(\kappa) excursions in the unit disk, starting from the boundary, growing towards all inner points simultaneously, invariant under all conformal self-maps of the disk. We prove that this conformal growth process of excursions, abbreviated as CGE(\kappa), exists iff \kappa\in [0,4), and that it does not create additional fractalness: the Hausdorff dimension of the closure of all the SLE(\kappa) arcs attached is 1+\kappa/8 almost surely. We determine the dimension of points that are approached by CGE(\kappa) at an atypical rate, and construct conformally invariant random fields on the disk based on CGE(\kappa).