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Spectral Radii of Products of Random Rectangular Matrices

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Abstract

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.

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Acknowledgements

The authors would like to thank an anonymous referee for his/her careful reading of the manuscript. The research of Yongcheng Qi was supported in part by NSF Grant DMS-1916014.

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  1. Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN, 55812, USA

    Yongcheng Qi & Mengzi Xie

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  1. Yongcheng Qi
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  2. Mengzi Xie
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Correspondence to Yongcheng Qi.

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Qi, Y., Xie, M. Spectral Radii of Products of Random Rectangular Matrices. J Theor Probab 33, 2185–2212 (2020). https://doi.org/10.1007/s10959-019-00942-9

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