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).

$$\begin{array}{c} n & P_n & Q_{n-1}& \\ 2 &1 & 1 \\ 3 &0.708 & 0.70711\\ 4 &0.346 & 0.35355\\ 5 &0.117 & 0.125\\ 6 & 0.028 & 0.03132\\ \end{array}$$

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 $P_n \approx Q_{n-1}$

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 $P_n \approx Q_{n-1}$

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).

$$\begin{array}{c} n & P_n & Q_{n-1}& \\ 2 &1 & 1 \\ 3 &0.708 & 0.70711\\ 4 &0.346 & 0.35355\\ 5 &0.117 & 0.125\\ 6 & 0.028 & 0.03132\\ \end{array}$$

Values of $P_n$ 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 $P_n \approx Q_{n-1}$

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).

$$\begin{array}{c} n & P_n & Q_{n-1}& \\ 2 &1 & 1 \\ 3 &0.708 & 0.70711\\ 4 &0.346 & 0.35355\\ 5 &0.117 & 0.125\\ 6 & 0.028 & 0.03132\\ \end{array}$$

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 $P_n \approx Q_{n-1}$