I am interested in finding the expected value of the product of two eigenvalue with respected to the GOE ensemble.

So I have a random $n\times n$ matrix with all off diagonal elements $\mathcal{N}(0,1)$ and on the diagonal $\mathcal{N}(0,2)$ all independent; it is well know that it is possible to evaluate the explicit p.d.f. for the eigenvalues of this kind of matrix: $$ f(\lambda)d\lambda = n! \Delta(\lambda)e^{-\frac 1 2\sum \lambda_j^2}$$

But with this formulation I am now able to compute for instance neither $\mathbb{E}[\lambda_j]$ or $\mathbb{E}[\lambda_j\lambda_i]$, there are any technique to evaluate those integrals? It is possible to use the asymptotic density function (i.e. the semicircle law) to compute them or at least to have an estimate with a remainder in terms of the size of the matrix?


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