论文标题
方向性的极端统计数据
Directional Extremal Statistics for Ginibre Eigenvalues
论文作者
论文摘要
我们考虑了大尺寸真实或复杂的Ginibre矩阵的特征值，其真实部分达到其最大值。该最大值遵循牙龈分布，这些极端特征值构成了泊松点过程，因为尺寸倾向于无穷大。在复杂的情况下，这些事实已经由Bender \ cite {MR2594353}建立，而在实际情况下，Akemann和Phillips \ cite {MR3192169}即使是对更一般的椭圆机的整体进行了复杂的鞍点分析。本说明的目的是在Ginibre情况下以有效的错误项给出非常短的直接证明。此外，我们对此制度中相关内核的估算是准确定位$ \ max \ re \ mathrm {spec}（x）$的关键输入。伴侣纸中的条目\ cite {2206.04448}。
We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process, asymptotically as the dimension tends to infinity. In the complex case these facts have already been established by Bender \cite{MR2594353} and in the real case by Akemann and Phillips \cite{MR3192169} even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this note is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating $\max\Re\mathrm{Spec}(X)$ for any large matrix $X$ with i.i.d. entries in the companion paper \cite{2206.04448}.
