The Resultant Distribution Method: Universality for $p$-adic Random Matrices and Polynomials

[2608.06576] The Resultant Distribution Method: Universality for $p$-adic Random Matrices and Polynomials Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2608.06576 (math) [Submitted on 6 Aug 2026 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: The Resultant Distribution Method: Universality for $p$-adic Random Matrices and Polynomials Authors: Jiahe Shen View PDF HTML (experimental) Abstract: We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski ( arXiv:2601.06283 ), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso ( arXiv:2110.03942 ) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions. Comments: 46 pages. Major revision with substantial reorganization and rewriting throughout. The only substantive technical update is the addition of the missing moment-tightness argument for random matrices, completing the random matrix eigenvalue universality theorem. All other changes are organizational or expository. Comments welcome! Subjects: Probability (math.PR) ; Number Theory (math.NT) Cite as: arXiv:2608.06576 [math.PR] (or arXiv:2608.06576v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2608.06576 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jiahe Shen [ view email ] [v1] Thu, 6 Aug 2026 20:38:06 UTC (73 KB) [v2] Mon, 24 Aug 2026 23:51:40 UTC (76 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.NT 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. 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