CAREER: Littlewood-Offord Theory and Universality in Random Structures

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Random systems are often difficult to analyze, but in many cases there often occurs a striking phenomenon known as universality, where many statistics of the systems are independent of the distributions of the components. Famous examples include the bell curve that appears in countless empirical histograms, or Benford's law that governs the first digit of many real-life sets of numerical data. While these universal laws are very well studied, there are numerous mysterious laws that are frequently observed, but not at all understood, especially those arising from random systems with complicated component correlations. A major part of this research project provides rigorous mathematical methods to discover and justify universality phenomena for various complex systems, with a special focus on random matrices and random polynomials. This study is expected to lead to a more complete and deeper understanding of these systems, with considerable impact on related areas of science, including mathematical physics, combinatorics, number theory, statistics, and theoretical computer science. The principal investigator will also run a number of seminars and workshops to help postdoctoral researchers, graduate students, and undergraduates in their professional career development, as well as to stimulate interaction across fields including, but not limited to, combinatorics and probability. In technical terms, the research project will develop novel methods to characterize inhomogeneous random walks of large returning probability in both discrete and continuous settings for non-abelian groups. This task also includes finding optimal characterizations of random multilinear forms with large concentration probability. These time-varying models of random walks are highly challenging because of their inhomogenity, but they appear very frequently in a number of probabilistic models; a systematic study of these walks is expected to have substantial impact. With respect to the universality phenomenon, the principal investigator will focus on random polynomials, random eigenfunctions of smooth manifolds, and random matrices. More specifically, the PI will study correlations of roots of random polynomials, nodal statistics of the random wave model with Bernoulli coefficients, and important questions involving the smallest singular values, the spectral repulsion, and the logarithmic determinant and permanent of random matrices of different types of symmetry and sparsity.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.