Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910051769253888 |
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| author | Jung, Jiwan Lee, Jungin Yu, Myungjun |
| author_facet | Jung, Jiwan Lee, Jungin Yu, Myungjun |
| contents | We prove that $\frac{\log n}{n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{Z}_p$. More precisely, let $α_n = \frac{c\log n}{n}$ for a constant $c>0$ and let $A(n)$ be an $α_n$-balanced random matrix over $\mathbb{Z}_p$. For non-symmetric, symmetric, and alternating matrix models, we prove that if $c>1$, then the limiting distribution of the cokernel of $A(n)$ coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale $c=1$. This improves earlier universality results, which required $α_n \gg \frac{\log n}{n}$, to the optimal threshold. As an application, we generalize the universality result for Sylow $p$-subgroups of sandpile groups of Erdős-Rényi random graphs to a broader class of Erdős-Rényi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers Jung, Jiwan Lee, Jungin Yu, Myungjun Combinatorics Number Theory Probability We prove that $\frac{\log n}{n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{Z}_p$. More precisely, let $α_n = \frac{c\log n}{n}$ for a constant $c>0$ and let $A(n)$ be an $α_n$-balanced random matrix over $\mathbb{Z}_p$. For non-symmetric, symmetric, and alternating matrix models, we prove that if $c>1$, then the limiting distribution of the cokernel of $A(n)$ coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale $c=1$. This improves earlier universality results, which required $α_n \gg \frac{\log n}{n}$, to the optimal threshold. As an application, we generalize the universality result for Sylow $p$-subgroups of sandpile groups of Erdős-Rényi random graphs to a broader class of Erdős-Rényi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately. |
| title | Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers |
| topic | Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2603.12879 |