Limiting distributions of spectral radii for product of matrices from the spherical ensemble. (English) Zbl 1385.15004
Summary: Consider the product of \(m\) independent \(n \times n\) random matrices from the spherical ensemble for \(m \geq 1\). The spectral radius is defined as the maximum absolute value of the \(n\) eigenvalues of the product matrix. When \(m = 1\), the limiting distribution for the spectral radii has been obtained by T. Jiang and Y. Qi [J. Theor. Probab. 30, No. 1, 326–364 (2017; Zbl 1362.15024)]. In this paper, we investigate the limiting distributions for the spectral radii in general. When \(m\) is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent gamma random variables. When \(m = m_n\) tends to infinity as \(n\) goes to infinity, we show that the logarithmic spectral radii have a normal limit.
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 60B20 | Random matrices (probabilistic aspects) |
Keywords:
limiting distribution; spectral radius; spherical ensemble; product ensemble; random matrixCitations:
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