Simple CLE in Doubly Connected Domains

TL;DR

This paper extends the construction of simple Conformal Loop Ensemble (CLE) to doubly connected domains like annuli and punctured surfaces, analyzing their properties and associated exploration processes.

Contribution

It introduces constructions of simple CLE in annuli, punctured discs, and punctured planes, conditioned on specific events, expanding the understanding of CLE in complex domains.

Findings

Constructed simple CLE in annuli, punctured discs, and punctured planes.

Analyzed properties of CLE in these domains, including gasket measures.

Developed exploration processes for each CLE variant.

Abstract

We study Conformal Loop Ensemble (CLE$_{\kappa}$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLE$_{\kappa}$ for which the loops are simple, i.e. $\kappa\in (8/3,4]$. In the paper "Conformal Loop Ensemble" by Sheffield and Werner, simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLEs, along with the corresponding exploration processes.