This project focuses on the application of techniques from probability and complex analysis to mathematical problems concerning scaling limits of random planar growth and lattice model that originate in physics. The main focus is on models related to Loewner's differential equation, and in particular on the Schramm-Loewner evolution (SLE). Three principal directions of investigation are proposed. First, the PI intends to study almost sure fine properties of the SLE curves, primarily from the point of view of multifractal analysis. One of the main questions concerns the properties of the almost sure multifractal spectrum of harmonic measure on the boundary of the SLE hulls, but the PI will also investigate, e.g., the winding of the curve at the tip and the geometry of the SLE curve's collisions with the boundary of the domain where it is defined. All of these questions can be formulated in terms of the boundary behavior of the random conformal maps that generate the SLE curves. Secondly, the PI intends to study rigorous connections between the SLE processes and related discrete models and Conformal Field Theory (CFT). One goal is to develop the rigorous understanding of the objects and algebraic structures of CFT from probabilistic and analytic points of view. Finally, the PI will study the Hastings-Levitov family of models which uses iterated random conformal maps to model planar aggregation processes. A specific regularization will be studied with an understanding of scaling limits and the predicted phase transition as ultimate goals.
Discrete probabilistic lattice models are used in physics to model a range of phenomena, and they provide a source of many interesting mathematical problems. Techniques related to the random fractal Schramm-Loewner evolution (SLE) curves, which approximate interfaces between phases in the models, have led to much progress in the mathematical understanding of such models in recent years. Within the project the PI will develop the understanding of the SLE curves' rich random geometric and multifractal structures. Beside the intrinsic and fundamental interest of these structures, and their strong connections with physical models, the universal nature of the SLE process suggests that methods and insights developed in the study of its geometric properties will be useful in the study of other random fractals. An important physics approach to lattice models uses Conformal Field Theories, CFTs, which (roughly) are ``smeared out'' continuum versions of the discrete models. The rigorous SLE machinery is still limited compared to the scope of the non-rigorous CFT predictions, and many connections between CFT and discrete models remain mysterious from a mathematical perspective. The PI intends to develop the rigorous understanding of the objects and structures of CFT from probabilistic and analytic points of view, in particular direct connections with SLE and the discrete models themselves. A related circle of questions concerns models of aggregation where planar clusters are grown to model important natural processes such as diffusion limited aggregation and flow of viscous fluid. Recent advances in the understanding of random growth models related to conformal maps has provided new tools which the PI will use to study versions of the so-called Hastings-Levitov family of aggregation models.
