I am looking for pointers/references to results of the following kind:

For $M$ a real or integer square matrix drawn at random from some "reasonably" nice set of square matrices (possibly infinite/uncountable), with a "reasonably" nice probability distribution, then with "reasonably" high probability the spectrum of $M$ either has exactly two dominant conjugate eigenvalues, or exactly one dominant real eigenvalue. And in either case (with high probability) these eigenvalues have algebraic multiplicity 1.

(By "dominant" I mean of modulus strictly larger than the other eigenvalues.)