Free Multiplicative Convolution and Erlang Moments in Monitored Quantum Transport

We study the transmission eigenvalues of monitored Haar products $$B_L=(P{S}_L)(P{S}_{L-1})\cdots (P{S}_1),$$ 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 $B_L^{†}{B}_L$ converges to ${\nu }_c^{⊠L}$, where ${\nu }_c=(1-c){\delta }_1+c{\delta }_0$. We then take the free small-loss limit and identify the limiting law by $$S_{{\mu }_{\tau }}(z)=\exp \left(\frac{\tau }{1+z}\right)\mathrm{.}$$ 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 $L\sim \tau N$, $c=1\mathrm{/}N$, as a quantitative convergence problem supported by low-order moment checks.