Free Multiplicative Convolution and Erlang Moments in Monitored Quantum Transport

We study the transmission eigenvalues of monitored Haar products $$B_L=(PS_L)(PS_{L-1})\cdots(PS_1),$$ where the $S_i$ are independent Haar unitaries and $P$ is a deterministic projection. For fixed $L$, we prove that the empirical eigenvalue distribution of $B_L^\dagger B_L$ converges to $\nu_c^{\boxtimes L}$, where $\nu_c=(1-c)\delta_1+c\delta_0$. We then take the free small-loss limit and identify the limiting law by $$S_{\mu_\tau}(z)=\exp\left(\frac{\tau}{1+z}\right).$$ Lagrange inversion gives explicit Erlang-type moments, explaining the polynomials appearing in Beenakker's recursion. We also record spectral consequences, including the atom $\mu_\tau(\{1\})=(1-\tau)_+$ and the real branch point $\tau \mathrm{e}^{1-\tau}$, and formulate the diagonal scaling $L\sim\tau N$, $c=1/N$, as a quantitative convergence problem supported by low-order moment checks.