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Title of the thesis: Large devivations of multiple SLE curves and related processes, and revised Prokhorov Le-Cam's theorem

Thesis defender: Osama Abuzaid 
Opponent: Professor Fredrik Viklund, KTH Royal Institute of Technology, Sweden
Custos: Associate Professor Eveliina Peltola, Aalto University School of Science

This thesis develops the large deviation theory of multichordal and multiradial Schramm-
Loewner evolution SLE(κ) and related processes as κ → 0. In addition, the thesis classifies sequences of random variables and random curves that have weakly converging subsequences.

SLE(κ) is a one-parameter family of random curves appearing in statistical physics, which in the limit κ → 0 approaches a deterministic curve. Multiradial and multichordal SLE(κ) consists of multiple interacting SLE(κ) curves. The thesis establishes a large deviation principle (LDP) for multiradial and multichordal SLE(κ) and the related Dyson-type diffusions. The LDP encodes the exponential rate of convergence into a rate function, which quantifies probabilities of extremely rare events. The precise formulation of LDP for SLEs established in the thesis is stronger than earlier LDP results. The LDP for Dyson type diffusions extends the LDP theory of Itô diffusions developed by Freidlin and Wentzell.

A random variable represents a random event, such as the result of a die roll or a realization of the aforementioned SLE(κ) curve. For special values of κ, the SLE(κ) curve is realized as the scaling limit of a critical lattice model coming from statistical physics. Scaling limit is defined in terms of weak convergence of a sequence of random variables. The Prokhorov-Le Cam's theorem mentioned in the title of the thesis states that so-called asymptotic tightness guarantees weak convergence along subsequences. This thesis shows that assumptions in asymptotic tightness can be slightly relaxed. The proof in the thesis is short and elementary compared to existing proofs of the Prokhorov-Le Cam's theorem. Furthermore, the thesis exactly characterizes sequences of random curves admitting subsequential weak limits.

Although the thesis is theoretical, the developed methods could be applied in more practical settings. For example, the thesis improves some general tools in the theory of large deviations, which have applications e.g. in risk theory and thermodynamics.

Keywords: Schramm-Loewner evolution, large deviation principle, Dyson-type diffusion,
precompactness, random curves

Thesis available for public display 7 days prior to the defence at . 

Contact information: 
osama.abuzaid@aalto.fi