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I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have

$$P(s)=C_{N,K}\int...\int\delta\left(s-\sum_{j=1}^N\lambda_j^2\right)\delta\left(1-\sum_{j=1}^N\lambda_j\right)\prod_i\lambda_i^{K-N}\prod_{i<j}(\lambda_i-\lambda_j)^2\prod_id\lambda_j$$

The limits being from $-\infty$ to $\infty$ for all the integrals and $N,K$ are constants. Now I tried expanding the Vandermonde determinant after getting rid of the delta functions and tried to use Selberg integral but unfortunately this is turning out to be very tedious and cumbersome. Even if its possible we'll end up sums of functions of gamma functions. Is there any other way to do this?

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  • $\begingroup$ the $\lambda$'s are complex in the Ginibre ensemble, because these are eigenvalues of a non-symmetric matrix; what does this integral mean? in particular this term $\delta(1-\sum_j\lambda_j)$ is zero. $\endgroup$
    – Carlo Beenakker
    Commented Nov 26, 2018 at 16:21
  • $\begingroup$ @CarloBeenakker My bad! I got confused. Its not Ginibre ensemble. But the joint distribution is correct i.e. $$P_{N,K}(\vec{\lambda})=\delta\left(1-\sum_{j=1}^N\lambda_j\right)\prod_i\lambda_i^{K-N}\prod_{i<j}(\lambda_i-\lambda_j)^2$$ I'm going through this. Pg2 $\endgroup$
    – Razor
    Commented Nov 26, 2018 at 17:26

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The expectation value of $s=\sum_{j}\lambda_j^2$ follows from equation 5.11 of arXiv:quant-ph/0405031, $$E(s)=\frac{K+N}{KN+1}.$$ The second moment is given by equation 5.16, and it is already a very lengthy expression.

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