Geometry and Integrability of Random Processes

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Two major goals of probability theory are to address the question of how large complex systems work and to identify the geometry of their evolution. Probabilistic models are widespread in fields like biology, statistical physics, quantum mechanics, and machine learning. Examples include models of cancer growth, spread of disease in population, governing principles of subatomic particles, black holes, neural networks, etc. The purpose of this project is to understand the geometry and intrinsic properties of a string of models that are representatives of these examples. The project aims to resolve open questions in those fields based on tools that the investigator has developed. Domino tilings, random matrices, and stochastic six vertex models are areas of intense interest in the field of statistical physics, while Liouville conformal field theory (LCFT) and theory of optimal transport have gained immense attention in the fields of quantum mechanics and machine learning. This project revolves around questions in those areas and aims to acquire new insights about their geometry and integrability. In particular, this project plans to: (1) find laws of iterated logarithms and fractal dimension of models in the Kardar-Parisi-Zhang (KPZ) universality class including the KPZ fixed point, edge spectrum of random matrices, and domino tilings; (2) build a unified framework for studying the moment formulas of interacting particle systems and vertex models including the stochastic six vertex model; (3) rigorously prove modular transformation properties of conformal blocks of LCFT and partition functions from gauge theory; and (4) study the convergence of entropically regularized optimal transport to optimal transport when the regularization vanishes. By intermingling ideas from various fields including geometry of polymers, representation theory, Riemann-Hilbert techniques, quantum groups, and convex geometry, the investigator aims to resolve questions that were hard to tackle with other methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.