Exploring Random Geometry with the Gaussian Free Field
Thesis
This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum. Chapter 1 is an introduction to Schramm-Loewner evolutions (SLE). SLEs are the canonical family of non-self-intersecting, conformally invariant random curves with a domain-Markov property. The family is indexed by a parameter, usually denoted by κ, which controls the regularity of the curve. We give the definition of the SLEκ process, and summarise the proofs of some of its properties. We give particular attention to the Rohde-Schramm theorem which, in broad terms, tells us that an SLEκ is a curve. In Chapter 2 we introduce the Gaussian free field (GFF), a conformally invariant random surface with a domain-Markov property. We explain how to couple the GFF and an SLEκ process, in particular how a GFF can be unzipped along a reverse SLEκ to produce another GFF. We also look at how the GFF is used to define Liouville quantum gravity (LQG) surfaces, and how thick points of the GFF relate to the quantum gravity measure. Chapter 3 introduces a diffusion on LQG surfaces, the Liouville Brownian motion (LBM). The main goal of the chapter is to complete an estimate given by N. Berestycki, which gives an upper bound for the Hausdor dimension of times that a γ-LBM spends in α-thick points for γ, α ∈ [0, 2). We prove the corresponding, tight, lower bound. In Chapter 4 we give a new proof of the Rohde-Schramm theorem (which tells us that an SLEκ is a curve), which is valid for all values of κ except κ = 8. Our proof uses the coupling of the reverse SLEκ with the free boundary GFF to bound the derivative of the inverse of the Loewner flow close to the origin. Our knowledge of the structure of the GFF lets us find bounds which are tight enough to ensure continuity of the SLEκ trace.