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arXiv · 2004.09266

Commutators of random matrices from the unitary and orthogonal groups

Pedro H. S. Palheta·Marcelo R. Barbosa·Marcel Novaes
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Abstract

We investigate the statistical properties of $C=uvu^{-1}v^{-1}$, when $u$ and $v$ are independent random matrices, uniformly distributed with respect to the Haar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived for the average value of power sum symmetric functions of $C$, and also for products of the matrix elements of $C$, similar to Weingarten functions. The density of eigenvalues of $C$ is shown to become constant in the large-$N$ limit, and the first $N^{-1}$ correction is found.

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Pedro H. S. Palheta, Marcelo R. Barbosa, Marcel Novaes. 2020-04-20. Commutators of random matrices from the unitary and orthogonal groups. https://doi.org/10.1063/5.0041240

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