Permanents of matrix ensembles: computation, distribution, and geometry
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Quantum Physics arXiv:2602.10141 (quant-ph) [Submitted on 8 Feb 2026] Title:Permanents of matrix ensembles: computation, distribution, and geometry Authors:Igor Rivin View a PDF of the paper titled Permanents of matrix ensembles: computation, distribution, and geometry, by Igor Rivin View PDF HTML (experimental) Abstract:We report on a computational and experimental study of permanents. On the computational side, we use the GPU to greaatly accelerate the computation of permanents over $\mathbb{C},$ $\mathbb{R},$ $\mathbb{F}_p$ and $\mathbb{Q}.$ In particular, we use this to compute the permanents of DFT and Schur matrices far beyond the ranges hitherto known. On the experimental side, we present two new observations. First, for Haar-distributed unitary matrices~$U$, the permanent $\perm(U)$ follows a circularly-symmetric complex Gaussian distribution $\mathcal{CN}(0,\sigma^2)$ -- we confirm this via a number of tests for $n$ up to~23 with $50{,}000$ samples. The DFT matrix permanent is an extreme outlier for every prime $n\ge 7$. In contrast, for Haar-random \emph{orthogonal} matrices~$O$, the permanent $\perm(O)$ is approximately real Gaussian but with positive excess kurtosis that decays as~$O(1/n)$, indicating slower convergence. For matrices with Gaussian entries (GUE, GOE, Ginibre), the permanent follows an $\alpha$-stable distribution with stability index $\alpha\approx 1.0$--$1.4$, well below the Gaussian value $\alpha=2$. Secondly, we study the permanent along geodesics on the unitary group. For the geodesic from the identity to the $n$-cycle permutation matrix, we find a universal scaling function $f(t)=\frac{1}{n}\ln|\perm(\gamma(t))|$ that is independent of~$n$ in the large-$n$ limit, with a midpoint value \[ \perm(\gamma({\textstyle\frac12})) = (-1)^{(n-1)/2}\cdot 2e^{-n}\bigl(1+\tfrac{1}{3n}+O(n^{-2})\bigr) \] for odd~$n$ and zero for even~$n$. For the geodesic to the DFT matrix, the permanent recovers $10$--$40$ times above its valley minimum when $n$ is prime, but not when $n$ is composite -- a geodesic fingerprint of primality. Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR) MSC classes: 2020]{15A15, 81P68 (primary), 60B20, 11C20, 05A15 Cite as: arXiv:2602.10141 [quant-ph] (or arXiv:2602.10141v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.10141 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Igor Rivin [view email] [v1] Sun, 8 Feb 2026 22:31:42 UTC (781 KB) Full-text links: Access Paper: View a PDF of the paper titled Permanents of matrix ensembles: computation, distribution, and geometry, by Igor RivinView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: math math-ph math.CO math.MP math.PR References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)