  We present a non perturbative calculation technique providing the mixed
moments of the overlaps between the eigenvectors of two large quantum
Hamiltonians: $\hat{H}_0$ and $\hat{H}_0+\hat{W}$, where $\hat{H}_0$ is
deterministic and $\hat{W}$ is random. We apply this method to recover the
second order moments or Local Density Of States in the case of an arbitrary
fixed $\hat{H}_0$ and a Gaussian $\hat{W}$. Then we calculate the fourth order
moments of the overlaps in the same setting. Such quantities are crucial for
understanding the local dynamics of a large composite quantum system. In this
case, $\hat{H}_0$ is the sum of the Hamiltonians of the system subparts and
$\hat{W}$ is an interaction term. We test our predictions with numerical
simulations.
