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

  title={The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system},
  author={Jinho Baik and Thomas Bothner},
  journal={The Annals of Applied Probability},
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… 

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