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arXiv · 2305.20029

Random commuting matrices

John E. McCarthy
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Abstract

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as $n$ tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.

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John E. McCarthy. 2023-05-31. Random commuting matrices. https://arxiv.org/abs/2305.20029

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