# Extreme eigenvalues of real symmetric matrix with main diagonal variance twice non-diagonal

Main question

Suppose there exists a random real symmetric $N \times N$ matrix $A$ with the main diagonal elements distributed according to $\mathcal N(\mu = 0, \sigma^2 = 4N)$, while all non-diagonal elements are distributed according to $\mathcal N(0, 2N)$. What are the limits of the eigenvalue spectrum (i.e. the lower limit of the smallest eigenvalue and the upper limit of the largest eigenvalue)?

$A = X^T W + W^TX$ where each element of $X$ equals $\pm 1$ and each element of $W$ is independently drawn from $\mathcal N(0,1)$. Each element of $X^TW$ (and $W^TX$) is the sum of $N$ i.i.d. random variables such that each element is distributed according to $\mathcal N(0,N)$. Summing $X^T W$ and $W^T X$ will therefore result in the matrix as described in the main question above.
1. Wigner's semicircle law might provide a solution, though the elements of $A$ are not i.i.d. (esp. the difference in variance on the main diagonal)