TITLE: Probability that leading Eigenvalue is real
QUESTION [7 upvotes]: What is the probability that the leading Eigenvalue (largest real part) of a large i.i.d Gaussian (real) random matrix is real? To what will this probability converge in the limit of large size? 
Update: my numerical experiments find that the fraction of leading eigenvalues that are real drops with size but appears to saturate somewhere around 0.3. I calculated this for matrices up to N=8000, albeit for only 80 realizations each.

REPLY [6 votes]: These matrices are known as (real) Ginibre ensembles.
The following paper1 proves that the limit is actually zero, and not one:

"To make this explicit, we check here that the probability of the
  largest point in absolute value being real tends to zero."

They also show this to be true experimentally (diagram taken from their paper):

And adding regarding your assumption:

The striking feature is the so-called “Saturn effect,” based on which
  alone a person might be forgiven for having conjectured that the
  largest eigenvalue would be real, with probability one, as
  $n\to\infty$. Rather, the Saturn effect is a phenomenon which appears
  from plotting the eigenvalues of many matrices simultaneously.
  Eventually, the complex points overwhelm the $O(\sqrt{n})$ on the real
  line.

1 Extremal laws for the real Ginibre ensemble Brian Rider and Christopher D. Sinclair