the explicit expression (\[pg\]) for $P_M(\gamma)$ and applying the change of variables $\gamma=2\vert z\vert^2$, Eq. (\[pdem\]) can then be cast in the following form: $$\begin{gathered}
\label{p_M(X)int}
 {\mathcal{P}^{\mathrm{pf}}}_M(X) = \frac{1}{\Gamma(M/2)}\int\limits_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\\
  \times \int\limits_{-\infty}^{+\infty}dz\,|z|^{M-1}\,e^{i\omega X-z^2} \frac{\sqrt{i\omega}z\pi}{\sinh(\sqrt{i\omega}z\pi)}\,.\end{gathered}$$

As one can easily check, this expression is properly normalized to unity. It is also worth noting that the integrand of Eq. (\[p\_M(X)int\]) is an analytic function in $\omega$ except for the poles located on the upper part of the imaginary axis at $\omega_k=i(k/z)^2$, $k=1,2,\ldots,\infty$. This readily implies that ${\mathcal{P}^{\mathrm{pf}}}_M(X)=0$ at $X<0$ identically.

(0,0)![(Color online) The distribution of the rescaled complexness parameter for the picket-fence model at $M=1,5$ and 10 (top, middle and bottom, respectively). The analytical result (\[p\_M(X)int\]) is plotted in the solid line while the histograms correspond to numerics. Insets show the tail of the distribution in a semi-log scale.](q_distr2.eps "fig:")

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The details of the subsequent 