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

The real Ginibre ensemble consists of $n\times n$ real matrices $\mathbf{X}$ 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 $R_{n}=\max_{1\leq j\leq n}|z_{j}(\mathbf{X})|$ of the eigenvalues $z_{j}(\mathbf{X})\in \mathbb{C}$ of a real Ginibre matrix $\mathbf{X}$ follows a different limiting law (as $n\rightarrow \infty $) for $z_{j}(\mathbf{X})\in \mathbb{R}$ than for $z_{j}(\mathbf{X})\in \mathbb{C}\setminus \mathbb{R}$. Building on previous work by Rider and Sinclair (Ann. Appl. Probab. 24 (2014) 1621–1651) and Poplavskyi, Tribe and Zaboronski (Ann. Appl. Probab. 27 (2017) 1395–1413), we show that the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ 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 $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ and extend recent tail estimates in (Ann. Appl. Probab. 27 (2017) 1395–1413) via nonlinear steepest descent techniques.