Wegner estimates and universality for non-Hermitian matrices

在概率论中，中心极限定理（CLT）解释了为什么任意分布的随机变量的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18世纪的de Moivre，但直到20世纪，CLT才得到严格的证明。在随机矩阵理论中也出现了类似的现象，称为普适性。在这种情况下，普适性猜想断言，大随机矩阵的特征值统计只依赖于矩阵的对称性，而与控制其随机行为的精确概率密度无关。此外，在大维度的限制下，特征值的分布就像从高斯分布中提取的条目一样。这一猜想类似于CLT，具有深刻的哲学和实践意义。在数值和实验中观察到，许多物理系统独立于其组成元素之间相互作用的精确细节而表现出相同的行为。随机矩阵的这一性质被推测为更普遍地成立。特别是，它实现了数学物理的中心目标之一：推导大系统的宏观性质，尽管未知或随机的相互作用的细节。到目前为止，它只在一些特殊的情况下得到了证明，比如维格纳合奏。本计画的目标是证明由非厄米随机矩阵组成的系综的各种特征值统计量的普适性。这样的集合在数学和物理文献中都有研究，尽管在各个研究领域都有广泛的应用，但还没有得到很好的理解。通过观察对厄米随机矩阵的功，我们可以假设哪些定理在非厄米随机矩阵的情况下是正确的。尽管该领域取得了缓慢而稳定的进展，但我们看到仍有几个困难和重要的问题有待解决。特别是，特征值相关函数的研究仍然远远落后于厄米相关函数。

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.