Explorations of Conformal Loop Ensembles on Deterministic and Random Surfaces

Abstract

The central topic of this thesis is the study of properties of Conformal Loop Ensembles (CLE), which are random conformally invariant collections of loops in the plane satisfying a spatial Markov property. A way to construct and analyze these CLE is by exploring the loops one after another, which gives rise to certain particular exploration mechanisms for CLEs. These explorations can be described in terms of variants of Schramm-Loewner-Evolutions (SLE) which allow us to deploy stochastic analysis tools. The first chapter is an introduction to the area of conformally invariant random objects and puts the various results of this thesis in a broader context. In the second chapter, we consider the fuzzy Potts model on a critical FK percolation with parameter smaller than 4, which is obtained by assigning i.i.d. colors (red or blue) to the FK percolation clusters. The chapter establishes a scaling limit result for this fuzzy Potts model and its critical behavior is analyzed. This is closely related to explorations of a CLE since it is conjectured (and in some cases known) that in the scaling limit, FK percolation cluster boundaries converge in distribution to a CLE with a certain parameter. The coloring of the clusters corresponds to uniquely specifying an exploration mechanism of the CLE. The third chapter deals with exploration procedures for CLEs with parameter 4; this can also be viewed as an analysis of the conjectured scaling limit of the fuzzy Potts model on a critical FK percolation with parameter 4. Here, the exploration procedure of the CLE is pinned down by a local time drift which we determine. In the final fourth chapter, we consider a CLE overlayed with a random environment. More precisely, we consider a Liouville Quantum Gravity (LQG) disk (which is a random planar geometry) and decorate it with a CLE. It is explained how to weight this decorated surface by the number of loops surrounding some points on the surface. This construction is related to certain natural discrete random planar map models and constructions from Liouville Conformal Field Theory (LCFT). The ideas in this chapter in particular lead to a new computation of the law of the conformal radius of a CLE loop using LQG techniques. By studying the topics above, we are naturally led to consider a wide variety of objects from probability theory in general (which are of independent interest). Among others, we will analyze the excursion theory of Bessel and Lévy processes, study the robustness of certain stochastic differential equations, perform particular Doob transforms of (variants of) Lévy processes and compute some asymptotics of renewal processes.