LEAPS-MPS: Analytic Approaches to Limit Theorems and Random Structures

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Large random structures are of central interest in mathematics and are used as models of various phenomena in physics, computer science, biology and many other branches of science. Some specific examples include models for magnetism, gases, the spread of tumors, and noise in data sets. The goal of this project is to advance the study of many of these important models through new analytic techniques. In addition to studying the specific models mentioned above, the project will build mathematical tools that can be used to study large random structures in general. The project will provide training opportunities that will involve undergraduate and graduate students in research on some of the proposed projects. This project centers on analytic approaches to problems in discrete probability. A central goal of the project is to create tools for demonstrating limit theorems, anti-concentration, and correlation decay for random variables based on analytic properties of generating and characteristic functions. These methods contain immediate applications to models in combinatorics and mathematical physics such as the Ising model, Gibbs point processes, and the hard core model. The PI will use similar complex- and Fourier-analytic techniques to study random algebraic structures such as random matrices, random polynomials, random partitions, and random characters of the symmetric group.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.