(7682,-2155)[(0,0)\[rb\]]{} (7682,-1711)[(0,0)\[rb\]]{} (5189,-1736)[(0,0)\[lb\]]{} (5189,-4134)[(0,0)\[lb\]]{} (4961,-3408)[(0,0)\[rb\]]{} (4961,-3000)[(0,0)\[rb\]]{} (4961,-2591)[(0,0)\[rb\]]{} (4961,-2184)[(0,0)\[rb\]]{} (4961,-1775)[(0,0)\[rb\]]{} (4961,-1367)[(0,0)\[rb\]]{}

A remark on the proportionality factor $f$ is appropriate here. In the RMT limit $N\to\infty$, this factor may be represented as follows $f=\frac{1}{2}\int_{0}^{\infty}\!ds\,s^{-2} R_2(s)$, where $R_2(s)$ is the two-point correlation function of the RMT. The main problem of the GOE case, already mentioned in [@Scho00; @Sav06], is an ‘infrared’ logarithmical divergency of $f$ due to $R_2(s)\sim s$ at $s \rightarrow 0$. Practically, this divergence can be regularized by introducing a cut-off at small $s$, $s\geq\epsilon$, see Fig. 1. Without this cut-off the expression of the complexness parameter obtained using first order perturbation theory (\[X\]) does not yield finite moments, thus demanding for the characterization of fluctuations of $X$ by means of its probability distribution.

Distribution function
=====================

The probability distribution function of the rescaled complexness parameter $X_n$ is defined as follows $$\label{PX}
  \mathcal{P}_M(X) = {\left \langle \delta(X-X_n) \right \rangle} \,,$$ where the statistical averaging over the levels, the widths and the projections is performed with the help of Eqs. (\[jpd\]), (\[pg\]), and (\[pZ\]), respectively. In the weak coupling regime, function (\[PX\]) depends only on the number $M$ of open channels.

It is instructive first to consider the case of the 