the picket-fence model have been performed. The closed Hamiltonian $H$ now belongs to GOE, its elements being defined by their first two moments: $${\left \langle H_{ij} \right \rangle}=0, \quad
{\left \langle H_{ij}^2 \right \rangle}=
    \begin{cases}
        4/(N \pi^2), & \; \; i=j \\
        2/(N \pi^2), & \; \; i\neq j,
    \end{cases}$$ where $N$ is the size of the matrix. Like in the picket-fence case, the normalization is chosen such that $\Delta=1/N$. Statistics were obtained with 150 matrices of size 1000$\times$1000. Only levels near $E=0$ for which spacings deviate less than 5% from $\Delta$ were kept. The agreement between numerical and analytical results is flawless, as shown in Fig. 3.

The comparison between the probability distribution of $X$ in the picket-fence model and for GOE illustrates the effects of the fluctuations of the spectrum on the complexness parameter. The maximum of both distributions are close to each other. This is mainly due to the spectrum rigidity in both ensembles. But at large $X$ the statistical weight is larger for GOE than for the picket-fence model. This difference is introduced by the behavior of the levels at small distance: the spacing of two eigenenergies can be very small, the corresponding contribution to the 