Probability that leading Eigenvalue is real

Question

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.

Answer 1

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}