On the solution of the Zakharov-Shabat system, which arises in the analysis of the largest real eigenvalue in the real Ginibre ensemble

Let ${\lambda }_{max}$ be a shifted maximal real eigenvalue of a random $N\times N$matrixwithindependent$N(0,1)$entries(the\mathrm{‘}realGinibrematri{x}^{\mathrm{\prime }})inthe$N\to\infty$ limit. It was shown by Poplavskyi, Tribe, Zaboronski \cite{PZT} that the limiting distribution of the maximal real eigenvalue has $s\to -\mathrm{\infty }$ asymptotics $$\mathbb{P}[{\lambda }_{max}<s]={\mathrm{e}}^{\frac{1}{2\sqrt{2\pi }}\zeta (\frac{3}{2})s+\mathcal{O}(1)},$$ where $\zeta$ is the Riemann zeta-function. This limiting distribution was expressed by Baik, Bothner \cite{BB18} in terms of the solution $q(x)$ of a certain Zakharov-Shabat inverse scattering problem, and the asymptotics was extended to the form $$\mathbb{P}[{\lambda }_{max}<s]={\mathrm{e}}^{\frac{1}{2\sqrt{2\pi }}\zeta (\frac{3}{2})t}c(1{+}_{\mathcal{O}}(1)),s\to -\mathrm{\infty }\mathrm{.}$$ We show that $q(x)$ is a smooth function, which behaves as $\frac{1}{x}$ as $x\to -\mathrm{\infty }\mathrm{.}$ Second, we show that the error term in the asymptotics is subexponential, i.e. smaller that ${\mathrm{e}}^{-C\mathrm{\mid }s\mathrm{\mid }}$ for any $C\mathrm{.}$ Third, we identify the constant $c$ as a conserved quantity of a certain fast decaying solution $u(x,t)$ of the Korteweg-de Vries equation. This, in principle, gives a way to determine $c$ via the known long-time $t\to +\mathrm{\infty }$ asymptotics of $u(x,t)\mathrm{.}$ We also conjecture a representation for the $c$ in terms of an integral of the Hastings-MacLeod solution of Painlev'e II equation.