论文标题
Wigner矩阵的边缘普遍性的一种新组合方法
A new combinatorial approach for edge universality of Wigner matrices
论文作者
论文摘要
在本文中，我们介绍了一种新的组合方法，以分析Wigner矩阵的巨大力量。我们的方法是由\ citet {sosh}从论文中激励的。但是计数方法是不同的。我们从类似于\ citet {az05}的经典单词句方法开始，然后从\ citet {sinaiSosh}，\ citet {sosh}和\ citet {peche2009universality}中获取动机，以将单词编码为类似于dyck路径的对象。确切地说，地图将一个单词带到戴克路径上，并从中删除了一些边缘。使用这种新计数，我们证明了具有次高斯条目的大型Wigner矩阵的边缘通用性。这种方法的一种新颖性与\ citet {sinaiSosh}，\ citet {sosh}和\ citet {peche2009 Universality}我们无需假设矩阵的条目对称分布在$ 0 $ $ 0 $左右。本文的主要技术贡献是两个折叠。首先，我们制作了``贡献单词''的编码（对于定义，人们可能会查看wigner矩阵的\ ref {sec：word}），从而检索了边缘的普遍性。因此，这是最好的。因此，我们希望这种方法适用于随机矩阵中的许多其他场景。非常相似。对于没有确切的计算，但存在一些组合结构的模型可能很重要。
In this paper we introduce a new combinatorial approach to analyze the trace of large powers of Wigner matrices. Our approach is motivated from the paper by \citet{sosh}. However the counting approach is different. We start with classical word sentence approach similar to \citet{AZ05} and take the motivation from \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} to encode the words to objects similar to Dyck paths. To be precise the map takes a word to a Dyck path with some edges removed from it. Using this new counting we prove edge universality for large Wigner matrices with sub-Gaussian entries. One novelty of this approach is unlike \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} we do not need to assume the entries of the matrices are symmetrically distributed around $0$. The main technical contribution of this paper is two folded. Firstly we produce an encoding of the ``contributing words" (for definition one might look at Section \ref{sec:word}) of the Wigner matrix which retrieves the edge universality. Hence this is the best one can do. We hope this method will be applicable to many other scenarios in random matrices. Secondly in course of the paper we give a combinatorial description of the GOE Tracy Widom law. The explanation for GUE is very similar. This explanation might be important for the models where exact calculations are not available but some combinatorial structures are present.
