  It has been known since the pioneering paper of Mark Kac, that the
asymptotics of Fredholm determinants can be studied using probabilistic
methods. We demonstrate the efficacy of Kac' approach by studying the Fredholm
Pfaffian describing the statistics of both non-Hermitian random matrices and
annihilating Brownian motions. Namely, we establish the following two results.
Firstly, let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a
random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre
matrix'). Consider the limiting $N\rightarrow \infty$ distribution
$\mathbb{P}[\lambda_{max}<-L]$ of the shifted maximal real eigenvalue
$\lambda_{max}$. Then \[ \lim_{L\rightarrow \infty}
e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)L}
\mathbb{P}\left(\lambda_{max}<-L\right) =e^{C_e}, \] where $\zeta$ is the
Riemann zeta-function and \[ C_e=\frac{1}{2}\log
2+\frac{1}{4\pi}\sum_{n=1}^{\infty}\frac{1}{n}
\left(-\pi+\sum_{m=1}^{n-1}\frac{1}{\sqrt{m(n-m)}}\right). \] Secondly, let
$X_t^{(max)}$ be the position of the rightmost particle at time $t$ for a
system of annihilating Brownian motions (ABM's) started from every point of
$\mathbb{R}_{-}$. Then \[ \lim_{L\rightarrow \infty}
e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)L}
\mathbb{P}\left(\frac{X_{t}^{(max)}}{\sqrt{4t}}<-L\right) =e^{C_e}. \] These
statements are a sharp counterpart of our previous results improved by
computing the terms of order $L^{0}$ in the asymptotic expansion of the
corresponding Fredholm Pfaffian.
