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

Abstract

Let $\lambda_{max}$ be a shifted maximal real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix') in the $N\to\infty$ limit. It was shown by Poplavskyi, Tribe, Zaboronski \cite{PZT} that the limiting distribution of the maximal real eigenvalue has $s\to-\infty$ asymptotics $$\mathbb{P} [ \lambda_{max} < s ] = {\rm e}^{\frac{1}{2\sqrt{2\pi}} \zeta(\frac32)s + \mathcal{O}(1)},$$ 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 $$\mathbb{P} [ \lambda_{max} < s ] = {\rm e}^{\frac{1}{2\sqrt{2\pi}}\zeta(\frac32)t} c(1+_\mathcal{O}(1)),\ s\to-\infty.$$ We show that $q(x)$ is a smooth function, which behaves as $\frac{1}{x}$ as $x\to-\infty.$ Second, we show that the error term in the asymptotics is subexponential, i.e. smaller that ${\rm e}^{-C|s|}$ for any $C.$ 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 $t\to+\infty$ asymptotics of $u(x,t).$ We also conjecture a representation for the $c$ in terms of an integral of the Hastings-MacLeod solution of Painlev\'e II equation.