CAREER: Littlewood-Offord Theory and Universality in Random Structures

随机系统通常很难分析，但在许多情况下，经常会出现一种称为普适性的显著现象，其中系统的许多统计数据与组件的分布无关。著名的例子包括出现在无数经验直方图中的钟形曲线，或者控制许多现实生活中的数字数据集的第一位数的本福德定律。虽然这些普遍定律被研究得很好，但也有许多神秘的定律经常被观察到，但根本不被理解，特别是那些来自具有复杂分量关联的随机系统的定律。这个研究项目的主要部分提供了严格的数学方法来发现和证明各种复杂系统的普适性现象，特别关注随机矩阵和随机多项式。这项研究有望对这些系统有更全面和更深入的了解，对相关科学领域产生相当大的影响，包括数学物理、组合数学、数论、统计学和理论计算机科学。首席研究员还将举办一些研讨会和工作坊，帮助博士后研究人员、研究生和本科生发展他们的职业生涯，并促进包括但不限于组合学和概率学在内的各个领域的互动。从技术上讲，该研究项目将开发新的方法来刻画非阿贝尔群在离散和连续环境中具有大返回概率的非均匀随机游动。这项任务还包括寻找具有大集中概率的随机多线性形式的最优特征。这些随机游动的时变模型具有很高的挑战性，因为它们不均匀，但它们在一些概率模型中非常频繁地出现；对这些游动的系统研究有望产生实质性的影响。关于普适性现象，主要的研究者将集中在随机多项式、光滑流形的随机特征函数和随机矩阵。更具体地说，PI将研究随机多项式的根的相关性，具有伯努利系数的随机波模型的节点统计，以及涉及不同类型对称和稀疏的随机矩阵的最小奇异值、谱排斥、对数行列式和永久性的重要问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.