Large Deviations and Related Topics in Random Conformal Geometry

This thesis explores three models of random processes in the complex plane: Schramm–Loewner evolution, the Hastings-Levitov model, and Dyson Brownian motion. A common theme throughout the thesis is the large deviation principle (LDP), which gives rise to functionals, called rate functions, which have intrinsic connections with the geometry of the models. Paper A presents a proof of the LDP for chordal Schramm–Loewner evo- lution, SLE𝜅, in the upper half-plane, as 𝜅 → 0+, in the topology of locally uniform convergence. The Loewner energy functional controls large deviations and is shown to be a good rate function. Paper B studies large deviations of the Hastings-Levitov HL(0) model in the small-particle limit, i.e., when the number of particles tends to infinity and the one-particle capacity vanishes while their product remains constant. In partic- ular, the growing cluster of particles attached to the unit disk is described via Loewner evolution, and we prove the LDP for the corresponding family of driving measures, with the rate function equal to the relative entropy. The LDP at the level of conformal maps is obtained via the contraction principle and leads to an interesting minimization problem of finding a driving measure with minimal relative entropy that produces a given cluster shape. We show that the class of shapes generated by finite-entropy Loewner evolution contains all Weil-Petersson and Becker quasicircles, a non-simple curve, and a Jordan curve with a cusp. Paper C proposes a rigorous definition of Dyson Brownian motion on a rectifiable Jordan curve. We show that the process can be constructed for inverse temperatures 𝛽 ≥ 1, and that the transition probability function satisfies the Fokker–Planck–Kolmogorov equation. Under additional smoothness assumptions on the curve, we prove convergence to the stationary Coulomb gas distribution on the curve, study large deviations at low temperature, and derive a mean-field McKean–Vlasov equation in the hydrodynamical limit. Paper D defines Dyson Brownian motion on a circular arc and is complemen- tary to Paper C. The process exists for all 𝛽 > 0, and its transition probability function satisfies the Fokker–Planck–Kolmogorov equation with reflecting bound- ary conditions. The process is ergodic and its stationary distribution is given by the Coulomb gas density on the circular arc.