Universality of Random Matrices and Semiclassical Quantum Theory

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The eigenvalue correlation functions for the matrix ensemble continuously interpolating between the asymmetric real and symmetric real random matrices were evaluated and the asymptotic behavior in the limit of large matrix dimension was derived. The method of semiclassical diagrammatic expansion was applied to classically chaotic quantum systems and the universal properties were reproduced. The random matrix methods were applied to the theory of complex networks and the eigenvalue densities of the matrices describing the networks were shown to be analytically evaluated in the limit of large average degree.