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.
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)
Xingzhi Zhan's "Extremal eigenvalues of real symmetric matrices with entries in an interval" (SIAM J. Matrix Anal. Appl. Vol. 27, No. 3, pp. 851-860) provides a solution to some extent. The only issue is limiting the normal distribution, which does not provide an exact solution.