Cohen-Lenstra flag universality for random matrix products
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909736604008448 |
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| author | Huang, Yifeng Nguyen, Hoi H. Van Peski, Roger |
| author_facet | Huang, Yifeng Nguyen, Hoi H. Van Peski, Roger |
| contents | For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this flag converges universally, for any nondegenerate entry distribution, to the Cohen-Lenstra type measure which weights each flag inversely proportional to the size of its automorphism group. As a corollary, we prove universality of certain formulas for the limiting conditional distribution of $\mathrm{cok}(M_1M_2)$ given $\mathrm{cok}(M_1),\mathrm{cok}(M_2)$ in terms of Hall-Littlewood structure constants, which were previously obtained only for Haar matrices over $\mathbb{Z}_p$.
Our proofs combine the general technology of Sawin-Wood, matrix product moment computations following those of Nguyen-Van Peski, and the computation done previously for Haar $p$-adic matrices by Huang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohen-Lenstra flag universality for random matrix products Huang, Yifeng Nguyen, Hoi H. Van Peski, Roger Probability Combinatorics Number Theory For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this flag converges universally, for any nondegenerate entry distribution, to the Cohen-Lenstra type measure which weights each flag inversely proportional to the size of its automorphism group. As a corollary, we prove universality of certain formulas for the limiting conditional distribution of $\mathrm{cok}(M_1M_2)$ given $\mathrm{cok}(M_1),\mathrm{cok}(M_2)$ in terms of Hall-Littlewood structure constants, which were previously obtained only for Haar matrices over $\mathbb{Z}_p$. Our proofs combine the general technology of Sawin-Wood, matrix product moment computations following those of Nguyen-Van Peski, and the computation done previously for Haar $p$-adic matrices by Huang. |
| title | Cohen-Lenstra flag universality for random matrix products |
| topic | Probability Combinatorics Number Theory |
| url | https://arxiv.org/abs/2508.10127 |