Wegner estimates and universality for non-Hermitian matrices

在概率论中，中心极限定理（CLT）解释了为什么来自任何分布的随机变量的平均值的分布遵循高斯曲线。虽然第一次观察到这种行为是由于在世纪的de Moivre，但直到20世纪世纪，CLT才被严格证明。类似的现象出现在随机矩阵理论中，称为普适性。在这种背景下，普适性猜想断言，大型随机矩阵的特征值统计仅取决于矩阵的对称性，而与控制其随机行为的精确概率密度无关。此外，在大维数的限制下，特征值的分布就好像从高斯分布中提取的条目。这个猜想类似于CLT，具有深刻的哲学和实践后果。在数值计算和实验中可以观察到，许多物理系统表现出相同的行为，而与其组成元素之间相互作用的精确细节无关。随机矩阵的这一性质被证明更一般地成立。特别是，它实现了数学物理学的中心目标之一：推导大系统的宏观性质，尽管相互作用的未知或随机细节。本课题的目标是证明由非Hermitian随机矩阵组成的系综的各种特征值统计量的普适性。这样的合奏已经在数学和物理文献中研究，尽管在各个研究领域的深远应用，并没有很好地理解。通过研究埃尔米特随机矩阵的工作，我们可以假设哪些定理在非埃尔米特情况下应该是正确的。尽管这一领域取得了缓慢和稳步的进展，但我们看到，若干困难和重要的问题仍有待解决。特别是，特征值相关函数的研究仍然远远福尔斯于厄米特对应。

In the theory of probability the Central Limit Theorem (CLT) explains why the distribution of the mean of random variables from any distribution follows the Gaussian curve. Although the first observation of this behaviour is due to de Moivre in the 18th century, it was only in the 20th century that the CLT was rigorously proven. A similar phenomenon appears in the theory of random matrices and is called universality. In this context, the universality conjecture asserts that the eigenvalue statistics of large random matrices depend only on the symmetries of the matrices, and are independent of the precise probability densities that govern their stochastic behavior. Furthermore, in the limit of large dimension, the eigenvalues are distributed as if the entries were drawn from the Gaussian distribution. This conjecture is akin to the CLT and has deep philosophical and practical consequences. It is observed in numerics and experiments that many physical systems demonstrate the same behavior independently of the precise details of interactions among their constituent elements. This property of random matrices is conjectured to hold more generally. In particular, it fulfils one of the central objectives in mathematical physics: the derivation of macroscopic properties of large systems, despite unknown or random specifics of interactions. So far, it has been proven only in a few specific cases, such as the Wigner ensemble.The goal of this project is to prove universality of various eigenvalue statistics for ensembles consisting of non-Hermitian random matrices. Such ensembles have been studied in both mathematics and physics literature and despite far-reaching applications in various fields of study, are not very well understood. By looking at the work on Hermitian random matrices, we can postulate which theorems should be true in the non-Hermitian case. Despite the slow and steady progress in the area, we see that several difficult and important problems yet remain to be solved. In particular, the study of eigenvalue correlation functions still falls far behind its Hermitian counterpart.