The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov-Shabat system

Understand

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 $z_j({\bf X})\in\mathbb{C}\setminus\mathbb{R}$.