实Ginibre系综的最大实本征值及其与Zakharov-Shabat系统的关系
The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov-Shabat system
摘要：
实Ginibre系综由$n\times n$实矩阵${\bf X}$组成，其元素是i.i.d.。标准正态随机变量与复Ginibre系综和四元数Ginibre系综形成鲜明对比的是，实Ginibre系综中的实特征值达到正似然。进而，实吉尼布尔矩阵${\bf X}$的本征值$z_j({\bf X})\in\mathbb{C}$的谱半径$R_n=\max_{1\leq j\leq n}|z_j({\bf X})|$对于$z_j({\bf X})\in\mathbb{R}$遵循与对于$z_j({\bf X})\in\mathbb{C}\setminus\mathbb{R}$不同的极限定律（如$n\rightarrow\infty$）。建立在以前的工作由骑士，辛克莱\cite{RS}和Poplavskyi，部落，Zaboronski \cite{PTZ}，我们表明，极限分布的$\max_{j:z_j\in\mathbb{R}}z_j({\bf X})$承认一个封闭的形式表达的一个杰出的解决方案的Zakharov-Shabat系统的逆散射问题。作为我们的分析的副产品，我们还获得了一个新的行列式表示的极限分布的$\max_{j:z_j\in\mathbb{R}}z_j({\bf X})$和最近的尾部估计在\cite{PTZ}通过非线性最速下降技术。
英文摘要：
The real Ginibre ensemble consists of $n\times n$ real matrices ${\bf X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_n=\max_{1\leq j\leq n}|z_j({\bf X})|$ of the eigenvalues $z_j({\bf X})\in\mathbb{C}$ of a real Ginibre matrix ${\bf X}$ follows a different limiting law (as $n\rightarrow\infty$) for $z_j({\bf X})\in\mathbb{R}$ than for $z_j({\bf X})\in\mathbb{C}\setminus\mathbb{R}$. Building on previous work by Rider, Sinclair \cite{RS} and Poplavskyi, Tribe, Zaboronski \cite{PTZ}, we show that the limiting distribution of $\max_{j:z_j\in\mathbb{R}}z_j({\bf X})$ admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. As byproducts of our analysis we also obtain a new determinantal representation for the limiting distribution of $\max_{j:z_j\in\mathbb{R}}z_j({\bf X})$ and extend recent tail estimates in \cite{PTZ} via nonlinear steepest descent techniques.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
