Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections

[2402.14631] Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Complex Variables arXiv:2402.14631 (math) [Submitted on 22 Feb 2024 ( v1 ), last revised 20 Jul 2026 (this version, v12)] Title: Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections Authors: Ozan Günyüz View a PDF of the paper titled Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections, by Ozan G\"uny\"uz View PDF HTML (experimental) Abstract: This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact Kähler manifolds and random polynomials on $\mathbb{C}^{m}$ in the setting of the weighted pluripotential theory. For random polynomials, we consider non-orthonormal bases and prove an equidistribution result which is more general than the ones acquired before for non-discrete probability measures. More precisely, our result demonstrates that the equidistribution holds true even when the random coefficients in the basis representation are not independent and identically distributed (i.i.d.), and moreover, they are not constrained to any particular probability distribution. For random holomorphic sections, by extending the concept of a sequence of asymptotically Bernstein-Markov measures introduced by Bayraktar, Bloom and Levenberg in their recent paper to the setting of holomorphic line bundles over compact Kahler manifolds, we derive a global equidistribution, variance estimate and expected distribution theorems related to the zeros of systems of random holomorphic sections for large tensor powers of a fixed holomorphic line bundle for any codimension k, generalizing a previous result of Bayraktar in his 2016 paper and giving also a positive answer to a question posed in the same paper, asking whether the equidistribution is true for non-homogeneous manifolds. For both random holomorphic polynomials on $\mathbb{C}^{m}$ and systems of random holomorphic sections, the variance estimation method detailed in another paper of the author with Bojnik is significant. Comments: A minor correction in Section 6, one reference updated, 20 pages Subjects: Complex Variables (math.CV) ; Differential Geometry (math.DG); Probability (math.PR) Cite as: arXiv:2402.14631 [math.CV] (or arXiv:2402.14631v12 [math.CV] for this version) https://doi.org/10.48550/arXiv.2402.14631 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Bulletin of the Korean Mathematical Society (2025) Submission history From: Ozan Günyüz [ view email ] [v1] Thu, 22 Feb 2024 15:26:00 UTC (35 KB) [v2] Sat, 24 Feb 2024 20:38:59 UTC (35 KB) [v3] Sat, 2 Mar 2024 08:47:15 UTC (34 KB) [v4] Tue, 5 Mar 2024 07:31:40 UTC (34 KB) [v5] Mon, 11 Mar 2024 16:01:02 UTC (34 KB) [v6] Mon, 18 Mar 2024 12:41:04 UTC (35 KB) [v7] Mon, 26 May 2025 17:07:32 UTC (36 KB) [v8] Mon, 7 Jul 2025 07:45:20 UTC (36 KB) [v9] Thu, 13 Nov 2025 19:01:48 UTC (23 KB) [v10] Fri, 10 Apr 2026 05:08:58 UTC (23 KB) [v11] Sun, 26 Apr 2026 12:45:29 UTC (23 KB) [v12] Mon, 20 Jul 2026 12:59:54 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections, by Ozan G\"uny\"uz View PDF HTML (experimental) TeX Source view license Current browse context: math.CV < prev | next > new | recent | 2024-02 Change to browse by: math math.DG math.PR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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