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Gaussian fluctuations for products of random matrices
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 144, Number 2, April 2022
- pp. 287-393
- 10.1353/ajm.2022.0006
- Article
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abstract:
We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N\times N$ random matrix ensembles. As $N\to\infty$, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for $M$ fixed and has a white noise component for $M\to\infty$ jointly with $N$. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for which we prove a central limit theorem for global fluctuations via certain conditions on the generating functions. We apply our approach to a number of ensembles, including square roots of Wishart, Jacobi, and unitarily invariant positive definite matrices with fixed spectrum, using a detailed asymptotic analysis of multivariate Bessel functions to verify the necessary conditions.