Spectral Radii of Products of Random Rectangular Matrices
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Spectral Radii of Products of Random Rectangular Matrices Yongcheng Qi1
· Mengzi Xie1
Received: 11 February 2019 / Revised: 3 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
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. Keywords Spectral radius · Eigenvalue · Random rectangular matrix · Non-Hermitian random matrix Mathematics Subject Classification (2010) 15B52 · 60F99 · 60G70 · 62H10
1 Introduction Since Wishart’s [46] work on large covariance matrices in multivariate analysis, the study of random matrices has drawn much attention from mathematics and physics communities and has found applications in areas such as heavy-nuclei [45], condensed matter physics [7], number theory [33], wireless communications [18], and high dimensional statistics [25,29,30]. Bouchaud and Potters [11] provide a survey on applications in finance. The interested reader can find more references in the Oxford Handbook of Random Matrix Theory by Akemann et al. [3]. Random matrix theory studies the eigenvalues of random matrices, including the properties of the spectral radii and the empirical spectral distributions of the eigenval-
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Yongcheng Qi [email protected] Mengzi Xie [email protected]
1
Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812, USA
123
Journal of Theoretical Probability
ues. Tracy and Widom [40,41] show that the largest eigenvalues of the three Hermitian matrices (Gaussian orthogonal ensemble, Gaussian unitary ensemble and Gaussian symplectic ensemble) converge in distribution to some limits which are now known as Tracy–Widom laws. Subsequently, the Tracy–Widom laws have found more applications, see, e.g., Baik et al. [6], Tracy and Widom [42], Johansson [28], Johnstone [29,30] and Jiang [25]. The study of non-Hermitian matrices has also attracted attention in the literature. Theoretical results in this direction can be applied to quantum chromodynamics, chaotic quantum systems and growth processes, dissipative quantum maps and fractional quantum Hall effect. More applications can be found in Akemann et al. [3] and Haake [22]. In the stimulating work by Rider [37,38] and Rider and Sinclair [39], the spectral radii of the real, complex and symplectic Ginibre ensembles are investigated. It is shown that the spectral radius of the complex Ginibre ensemble converges to the Gumbel distribution. Jiang and Qi [26] study the largest radii of three
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