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.)



Spectrum of random matrices has dominant eigenvalues
The "good" matrices, those whose eigenvalues of largest absolute value are either a single simple real eigenvalue or a conjugate pair of simple eigenvalues, form a dense open set in the $n \times n$ real matrices. In fact, if $B$ is any "good" $n \times n$ matrix and $A$ any $n \times n$ matrix,
$A + t B$ will be "good" for almost every real $t$. So the "good" matrices form a set of full $n^2$-dimensional Lebesgue measure, and a 
random matrix from any absolutely continuous probability distribution will be "good" with probability $1$.


Insofar as, after scaling, a "reasonably nice" distribution of integer matrices approximates an absolutely continuous distribution, it should produce
"good" matrices with high probability.


