in terms of the angles $\theta_{np}$ between the pairs of these $N$ vectors, $$\Gamma_{np} = (\mathbf{V}_n\cdot\mathbf{V}_p) = \sqrt{\Gamma_n\Gamma_p}\cos \theta_{np}.$$ The main advantage of this representation is that the angles $\theta_{np}$ are mutually independent and also independent of $\Gamma_n$. The probability distribution of any angle (for $M\geq2$) can be easily found to be given by the expression for a solid angle in an $M$-dimensional space [@Sok89]: $$\label{pt}
  p_M(\theta)=\frac{\Gamma(M/2)}{\sqrt{\pi}\Gamma((M-1)/2)}\sin^{M-2}\theta .$$ Note that $\Gamma_{np}=\sqrt{\Gamma_n\Gamma_p}$ at $M=1$. As concerns the lengths of these vectors, i.e., the widths (\[gam\]), these are well-known to be independent and $\chi^2$ distributed according to $$\label{pg}
  P_M(\gamma) = \frac{1}{2^{M/2}\Gamma(M/2)}\gamma^{M/2-1}e^{-\gamma/2}\,.$$ Henceforth $\gamma_n=\Gamma_n/\sigma^2$ stands for the dimensionless widths. This distribution function has the mean value ${\left \langle \gamma \right \rangle}=M$ and the variance $$\label{gmg2m}
  \mathrm{var}(\gamma) = 2M = \frac{2}{M}{\left \langle \gamma \right \rangle}^2\,.$$ Thus the widths cease to fluctuate as the number of open channels grows, with the average width being kept fixed.

It is now convenient to express all the quantities in their natural units and to consider a rescaled complexness parameter $X_n$ defined as follows: $$\label{X}
  \quad X_n \equiv \frac{\Delta^2}{\sigma^4}q^2_n
  = \gamma_n\sum_{p\ne n}\frac{\Delta^2Z_p}{4(E_n-E_p)^2}\,,$$ where we have introduced the following quantities $$\label{Z}
 Z_p = \gamma_p\cos^2\theta_{np}\,.$$ $Z_p$ may be given a geometrical 