Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws

[2604.25785] Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematical Physics arXiv:2604.25785 (math-ph) [Submitted on 28 Apr 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws Authors: B.Shapiro View a PDF of the paper titled Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws, by B.Shapiro View PDF HTML (experimental) Abstract: We study the asymptotic distribution of level crossings for random matrix pencils A_n+\lambda B_n in several ensembles, including complex and real i.i.d. matrices and Gaussian/Hermitian settings. We derive a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and formulate conditions under which its limit is governed by a deterministic potential. Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, we prove convergence of the empirical measure of level crossings to an explicit deterministic limit. In the complex Gaussian case these assumptions are verified (modulo a uniformity step), while in the general i.i.d. setting the results are conditional and motivated by universality theory. We further analyze the real case, showing that any limiting measure does not concentrate on the real projective line under suitable hypotheses, and discuss analogous phenomena for elliptic/Hermitian ensembles. Our results highlight the role of logarithmic energy and universality in governing spectral degeneracies of random matrix pencils. Comments: 19 pages, 4 figures, several results strengthened Subjects: Mathematical Physics (math-ph) ; High Energy Physics - Theory (hep-th) MSC classes: 15B52, 81Q15 Cite as: arXiv:2604.25785 [math-ph] (or arXiv:2604.25785v2 [math-ph] for this version) https://doi.org/10.48550/arXiv.2604.25785 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Boris Shapiro [ view email ] [v1] Tue, 28 Apr 2026 15:53:12 UTC (1,394 KB) [v2] Sat, 15 Aug 2026 11:10:11 UTC (92 KB) Full-text links: Access Paper: View a PDF of the paper titled Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws, by B.Shapiro View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2026-04 Change to browse by: hep-th math math.MP References & Citations INSPIRE HEP NASA ADS Google 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? ) 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? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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