An optimal Poincaré inequality for the complex Ginibre log-gas

[2608.19358] An optimal Poincaré inequality for the complex Ginibre log-gas Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2608.19358 (math) [Submitted on 19 Aug 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: An optimal Poincaré inequality for the complex Ginibre log-gas Authors: Djalil Chafaï View PDF HTML (experimental) Abstract: We establish an optimal Poincaré inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. The proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the Hörmander-Berndtsson estimate. It is short and self-contained. It remains valid, beyond the quadratic potential, for rotationally invariant additive plurisubharmonic potential perturbations. Additionally, we provide four alternative proofs, two of which yield sum-of-squares formulas for the deficit, based respectively on a Hermite expansion and on an integrated Bochner-Kodaira formula, the other two use the spectral analysis of a two-sided number-operator factorization and a Hermite-Slater polynomial expansion. We furthermore present new results related to linear statistics, Gaussian factorization, polynomial eigenfunctions, log-Sobolev inequalities, matrix lift and eigenvector overlaps, and non-quadratic potentials. Comments: 30 pages, 2 graphics Subjects: Probability (math.PR) ; Mathematical Physics (math-ph); Functional Analysis (math.FA) MSC classes: Primary 60E15, 60B20, Secondary 60J60, 32W05, 82B21, 82C31 Cite as: arXiv:2608.19358 [math.PR] (or arXiv:2608.19358v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2608.19358 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Djalil Chafai [ view email ] [v1] Wed, 19 Aug 2026 18:20:55 UTC (309 KB) [v2] Thu, 27 Aug 2026 04:31:35 UTC (334 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-08 Change to browse by: math math-ph math.FA math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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