  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'). We study the large deviations behaviour of the limiting $N\rightarrow
\infty$ distribution $P[\lambda_{max}<t]$ of the shifted maximal real
eigenvalue $\lambda_{max}$. In particular, we prove that the right tail of this
distribution is Gaussian: for $t>0$, \[
P[\lambda_{max}<t]=1-\frac{1}{4}\mbox{erfc}(t)+O\left(e^{-2t^2}\right). \] This
is a rigorous confirmation of the corresponding result of Forrester and Nagao.
We also prove that the left tail is exponential: for $t<0$, \[
P[\lambda_{max}<t]=
e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)t+O(1)}, \] where $\zeta$
is the Riemann zeta-function.
  Our results have implications for interacting particle systems. The edge
scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a
rescaling of a fixed time distribution of annihilating Brownian motions (ABM's)
with the step initial condition. Therefore, the tail behaviour of the
distribution of $X_s^{(max)}$ - the position of the rightmost annihilating
particle at fixed time $s>0$ - can be read off from the corresponding answers
for $\lambda_{max}$ using $X_s^{(max)}\stackrel{D}{=} \sqrt{4s}\lambda_{max}$.
