Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $X_n$ with $|\lambda_1|\leq\dots\leq|\lambda_n|$. Let $p_n$ denote the probability that $|\lambda_n|>|\lambda_{n-1}|$ (in this case, $\lambda_n$ is real). Then is $\lim_{n\rightarrow\infty}p_n \rightarrow 0$? If so, then what are the asymptotics for $p_n$? Do we at least know that $\limsup_{n\rightarrow\infty}p_n<1$?
My computer calculations suggest that $p_n\rightarrow 0$.
