Random matrices having all real eigenvalues: uniform vs gaussian distributions

Question

Let $P_n$ be the probability that a $n \times n$ real random matrix with independent entries and uniformly distributed on $[0,1]$ has all real eigenvalues.

Let $Q_n$ be the same probability, for a standard normal distribution.

I've found, empirically (comments in this unanswered MSE question), that $P_n$ behaves quite similarly to $Q_{n-1}$ (at least for the small values of $n$ I tried).

$$\begin{array}{c} n & P_n & Q_{n}& \\ 1 & 1 & 1\\ 2 & 1 & 0.7071\\ 3 & 0.708 & 0.3536\\ 4 & 0.346 & 0.1250\\ 5 & 0.117 & 0.0313\\ 6 & 0.028 & 0.0007 \end{array}$$

Values of $P_n$ are approximate, empirical, from my simulations. Values of $Q_n=2^{-n(n-1)/4}$, from "The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law", A. Edelman, Journal of Multivariate Analysis, 60, 203-232 (1997)

I'd like to find out an expression for $P_n$, and/or some argument that helps to explain the approximation $P_n \approx Q_{n-1}$

Answer 1

Real eigenvalues of non-Gaussian random matrices and their products finds that the probability that all eigenvalues are real is larger for distributions with large weight at the origin and that decay slowly away from the origin. The dependence on the distribution vanishes quickly for larger $n$, see section 8 of How Many Eigenvalues of a Random Matrix are Real?