For a real matrix $A$, define its second eigenvalue to be $\max_{i\ge 2}|\lambda_i|$, where $\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_n$ are the eigenvalues of $A$.
What is the expectation of the second eigenvalue of a random $\{0,1\}$, $n×n$ matrix, where each entry of the matrix is a Bernoulli random variable with probability $p\in(0,1)$, independent of all other entries?
I am interested in known results for large $n$'s. Any reference might be helpful.