Ginibre特征值的方向极值统计量
Directional Extremal Statistics for Ginibre Eigenvalues
摘要：
本文研究了高维实或复Ginibre矩阵在其实部达到最大值的复平面区域内的特征值。该最大值遵循Gumbel分布，并且这些极值特征值形成泊松点过程，当维数趋于无穷大时渐近。在复杂情况下，这些事实已经由Bender \cite{MR2594353}建立，在实际情况下，甚至对于更一般的椭圆系综，也由Akemann和Phillips \cite{MR3192169}建立，并进行了复杂的鞍点分析。本说明的目的是给一个很短的直接证明在Ginibre的情况下与有效的错误项。此外，我们的估计在这个制度的相关内核作为一个关键的输入准确定位$\max\Re\mathrm{Spec}(X)$的任何大矩阵$X$与i.i.d.相关论文\cite{2206.04448}中的条目。
英文摘要：
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}.
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