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均值的分布遵循高斯曲线。尽管对这一行为的第一次观察是在18世纪的德·莫弗，但直到20世纪，CLT才得到了严格的证明。类似的现象出现在随机矩阵理论中，称为普适性。在此背景下，普适性猜想断言，大型随机矩阵的特征值统计量仅取决于矩阵的对称性，而与支配其随机行为的精确概率密度无关。此外，在大维的限制下，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后果。在数值计算和实验中观察到，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猜测为更一般地成立。特别是，它满足了数学物理的中心目标之一：尽管相互作用的未知或随机细节，但推导出大系统的宏观性质。到目前为止，它只在一些特定的情况下被证明，例如Wigner系综.这个项目的目的是证明由非厄米特随机矩阵组成的系综的各种特征值统计的普适性.这样的合奏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，但还没有被很好地理解。通过观察厄米特随机矩阵的工作，我们可以假设哪些定理在非厄米特情况下应该是真的。尽管该领域取得了缓慢和稳定的进展，但我们看到，仍有几个困难和重要的问题有待解决。特别是，本征值相关函数的研究仍然远远落后于厄米相关函数的研究。

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.