complexness parameter is large, then the tail of $\mathcal{P}_M(X)$ is larger for GOE than for the picket-fence model. This feature is most explicitly seen by comparing the corresponding limiting distributions at $M\gg1$. The distribution (\[p(x)\]) of $x$ is easily obtained from Eq. (\[pXgoe\]) and reads $$\label{p(x)goe}
 p_{\mathrm{goe}}(x) = \frac{\pi^2}{24x^2} \Big(1+\frac{\pi^2}{4x}\Big) \exp \left(-\frac{\pi^2}{8x}\right)$$ In contrast to the asymptotic exponential behavior in the picket-fence case, $p_{\mathrm{pf}}(x)\propto e^{-2x}$, see Eq. (\[p(x)pf\]), the tail of the distribution (\[p(x)goe\]) follows a power-law decay: $p_{\mathrm{goe}}(x) \propto x^{-2}$.

Conclusion
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In this paper, we have studied the statistics of complex wavefunctions associated to the resonances of weakly opened wave chaotic systems with the preserved time-reversal symmetry. More specifically, in the perturbative regime, we have considered the case of the completely rigid spectra defined through the picket-fence model and that of the GOE displaying spectral fluctuations. One of the key features of this study relies on the proportionality between the average of the complexness parameter and the variance of the resonance widths, which we believe is valid for generic nondegenerate spectra. We have also derived the exact probability distribution of the complexness parameter in these two cases.

To check the validity of the present results, recent experiments in 