{"paper":{"title":"Free Multiplicative Convolution and Erlang Moments in Monitored Quantum Transport","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP","quant-ph"],"primary_cat":"math.PR","authors_text":"Joon Hyung Lee","submitted_at":"2026-07-06T23:21:51Z","abstract_excerpt":"We study the transmission eigenvalues of monitored Haar products \\[\n  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 \\[\n  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 recurs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05693","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05693/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}