A random walk on the category of finite abelian $p$-groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929457338515456 |
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| author | Lvov, Nikita |
| author_facet | Lvov, Nikita |
| contents | We study an irreducible Markov chain on the category of finite abelian $p$-groups, whose stationary measure is the Cohen-Lenstra distribution. This Markov chain arises when one studies the cokernel of a random matrix $M$, after conditioning on a submatrix of $M$. We show two surprising facts about this Markov chain. Firstly, it is reversible. Hence, one may regard it is a random walk on finite abelian $p$-groups. The proof of reversibility also explains the appearance of the Cohen-Lenstra distribution in the context of random matrices. Secondly, we can explicitly determine the spectrum of the infinite transition matrix associated to this Markov chain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A random walk on the category of finite abelian $p$-groups Lvov, Nikita Probability Commutative Algebra Number Theory 15B52 We study an irreducible Markov chain on the category of finite abelian $p$-groups, whose stationary measure is the Cohen-Lenstra distribution. This Markov chain arises when one studies the cokernel of a random matrix $M$, after conditioning on a submatrix of $M$. We show two surprising facts about this Markov chain. Firstly, it is reversible. Hence, one may regard it is a random walk on finite abelian $p$-groups. The proof of reversibility also explains the appearance of the Cohen-Lenstra distribution in the context of random matrices. Secondly, we can explicitly determine the spectrum of the infinite transition matrix associated to this Markov chain. |
| title | A random walk on the category of finite abelian $p$-groups |
| topic | Probability Commutative Algebra Number Theory 15B52 |
| url | https://arxiv.org/abs/2408.06492 |