A phase transition for the cokernels of random band matrices over the p-adic integers

Andr\'as M\'esz\'aros

A phase transition for the cokernels of random band matrices over the p-adic integers

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arxiv 2408.13037 v1 pith:FGHNENCE submitted 2024-08-23 math.PR math.CO

András Mészáros This is my paper · ORCID

classification math.PRmath.CO

keywords bandinftymathbfcohen-lenstracokernelsdistributionhaar-uniformintegers

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abstract

Let $\mathbf{B}_n$ be an $n\times n$ Haar-uniform band matrix over $\mathbb{Z}_p$ with band width $w_n$. We prove that $\text{cok}(\mathbf{B}_n)$ has Cohen-Lenstra limiting distribution if and only if \[\lim_{n\to\infty} \left(w_n-\log_p(n)\right)=+\infty.\]

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The Cohen-Lenstra distribution for cokernels of random p-adic matrices persists when up to a linear fraction of entries per row and column are non-random, when within-column dependence is allowed, and for band matrice...

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