Spectral moments of complex and symplectic non-Hermitian random matrices

[2505.12271] Spectral moments of complex and symplectic non-Hermitian random matrices Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematical Physics arXiv:2505.12271 (math-ph) [Submitted on 18 May 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Spectral moments of complex and symplectic non-Hermitian random matrices Authors: Gernot Akemann , Sung-Soo Byun , Seungjoon Oh View a PDF of the paper titled Spectral moments of complex and symplectic non-Hermitian random matrices, by Gernot Akemann and 2 other authors View PDF HTML (experimental) Abstract: We study non-Hermitian random matrices belonging to the symmetry classes of the complex and symplectic Ginibre ensemble, and present a unifying and systematic framework for analysing mixed spectral moments involving both holomorphic and anti-holomorphic parts. For weight functions that induce a recurrence relation of the associated planar orthogonal polynomials, we derive explicit formulas for the spectral moments in terms of their orthogonal norms. This includes exactly solvable models such as the elliptic Ginibre ensemble and non-Hermitian Wishart matrices. In particular, we show that the holomorphic spectral moments of complex non-Hermitian random matrices coincide with those of their Hermitian limit up to a multiplicative constant, determined by the non-Hermiticity parameter. Moreover, we show that the spectral moments of the symplectic non-Hermitian ensemble admit a decomposition into two parts: one corresponding to the complex ensemble and the other constituting an explicit correction term. This structure closely parallels that found in the Hermitian setting, which naturally arises as the Hermitian limit of our results. Within this general framework, we perform a large-$N$ asymptotic analysis of the spectral moments for the elliptic Ginibre and non-Hermitian Wishart ensemble, revealing the mixed moments of the elliptic and non-Hermitian Marchenko--Pastur laws. Furthermore, for the elliptic Ginibre ensemble, we employ a recently developed differential operator method for the associated correlation kernel, to derive an alternative explicit formula for the spectral moments and obtain their genus-type large-$N$ expansion. Comments: 25 pages. Accepted for publication in J. Math. Phys Subjects: Mathematical Physics (math-ph) ; Probability (math.PR) Cite as: arXiv:2505.12271 [math-ph] (or arXiv:2505.12271v2 [math-ph] for this version) https://doi.org/10.48550/arXiv.2505.12271 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Seungjoon Oh [ view email ] [v1] Sun, 18 May 2025 07:21:33 UTC (30 KB) [v2] Thu, 9 Apr 2026 06:54:37 UTC (31 KB) Full-text links: Access Paper: View a PDF of the paper titled Spectral moments of complex and symplectic non-Hermitian random matrices, by Gernot Akemann and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2025-05 Change to browse by: math math.MP math.PR References & Citations 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

Related documents

• Rate of convergence for non-Hermitian random matrices and their products#779237
• Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices#1014702
• Spectral Universality for Matrices with Non-Linear Correlated Entries#1144960
• The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law#986349
• Phase transitions in non-Hermitian spherical integrals#1034485