calculations of ${\mathcal{P}^{\mathrm{pf}}}_M(X)$ are given in Appendix \[AppB\]. The final expression reads: $$\begin{gathered}
\label{p_M(X)bessel}
  {\mathcal{P}^{\mathrm{pf}}}_M(X) = \frac{2\pi (\sqrt{X})^{M/2-1}}{\Gamma(M/2)}\\ \times \int\limits_{0}^{\infty}dz\, \frac{z^{M/2+1}}{\sinh(z\pi)} J_{M/2-1}(2\sqrt{X}z) \,,\end{gathered}$$ with $J_\nu(x)$ being the Bessel function of order $\nu$. In the case of an odd number of channels, $M=2n+1$, $n=0,1,\ldots,$ this expression can be integrated further to yield an attractively simple formula $$\label{p_M(X)odd}
  {\mathcal{P}^{\mathrm{pf}}}_{2n+1}(X) = \frac{\sqrt{\pi}X^{n-1/2}}{2\Gamma(n+1/2)} \left(-\frac{\partial}{\partial X}\right)^{n} \frac{1}{\cosh^2(\sqrt{X})}\,.$$ In particular, the single-channel distribution ${\mathcal{P}^{\mathrm{pf}}}_1(X)$ reads $$\label{p_1(X)}
  {\mathcal{P}^{\mathrm{pf}}}_1(X) = \frac{1}{2\sqrt{X}}\frac{1}{\cosh^2(\sqrt{X})}\,.$$

It is interesting now to study in details the case of the large number of weakly open channels $M\gg1$. In view of the scaling (\[Xav\]), we consider the limiting probability distribution of $x=X/M$ defined as $$\label{p(x)}
 p(x) = \lim_{M\to\infty} M \mathcal{P}_M(Mx) \,.$$ Expression (\[p\_M(X)bessel\]) is actually not very convenient for evaluating this function. However, one can note that in the limit considered, the distribution $P_M(\gamma)$, Eq. (\[pg\]), tends to the Dirac distribution, $\delta(\gamma-M)$. Then, starting from Eq. (\[pdem\]), the integration over $\gamma$ is trivial and the probability distribution of $x$ reads: $$p_{\mathrm{pf}}(x) = \frac{1}{2\pi}\int\limits_{-\infty}^\infty d\omega \;e^{i\omega x}\prod_{k=1}^{\infty} \frac{1}{1+i\omega/(2k^2)}$$ Using the residue theorem, one readily gets: $$\label{p(x)pf}
 p_{\mathrm{pf}}(x) = 4\sum_{k=1}^{\infty}(-1)^{k+1} k^2 e^{- 2k^2x}$$ and finally $$\label{jacobi}
 p_{\mathrm{pf}}(x) = -2e^{-2x}\frac{d}{dx} \vartheta_4 (0,e^{- 2x})$$ where $\vartheta_4$ is a Jacobi theta 