$M$. On the other side, it captures fluctuation properties of the widths properly, e.g., yielding the vanishing ${\left \langle X \right \rangle}$ in the absorptive limit of many weakly coupled channels with the average total width kept fixed, due to the vanishing variance (\[gmg2m\]). Therefore, we believe that relation (\[Xvar\]) is a general feature of weakly open chaotic systems with non-degenerate spectrum in the perturbative regime. Figure 1 supports this suggestion through numerical simulations of the picket-fence and GOE models (with the details being given later in the next section).

(0,0)![The average rescaled complexness parameter versus the width variance for the GOE and picket-fence models. The symbols correspond to the results of numerical simulations performed at $M=1,2,3,5,10$ (see the text for details). The linear dependence predicted by Eq. (\[Xvar\]) is represented by the solid line. In the GOE case (a), the proportionality factor is given by the regularized expression $f_{\epsilon}=\frac{1}{2}\int_{\epsilon}^{\infty}\!ds\,s^{-2} R_2(s)$. The results obtained with three different values of the cut-off $\epsilon$ are shown. In the picket-fence case (b), $f=\pi^2/12$ as exactly given by Eq. (\[fpf\]).](q_distr1.eps "fig:")

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