Recent exact and asymptotic results for products of independent random matrices
Gernot Akemann, Jesper R. Ipsen
备注：
39 pages, review based on presentations at "Random Matrix Theory: Foundations and Applications", July 1-6, 2014, Krakow, Poland; v2: typo corrected, 2 clarifying comments added
发表信息：
Acta Physica Polonica B Vol. 46 No 9 (2015), 1747-1784
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.
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Asymptotic behavior of eigenvalues of large rank perturbations of large random matrices
Kyiv School of Economics（基輔經濟學院）；B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（烏克蘭國家科學院低溫物理與工程學院）；Department of Mathematics, Pennsylvania State University（賓夕法尼亚州立大學數學系）
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