Theory of Stochastic Canonical Equations of Random Matrix Physics, SOS Law, Elliptical Galactic Law, Sand Clock Law And Heart Law, Life, Sombrero and Halloween Laws

Our studies are essentially based on the martingale differences method developed in my previous papers for resolvents of random matrices. This method possesses the self-averaging property of the entries of resolvents of random matrices and, hence, we can deduce the stochastic canonical equation. The lecture contains the most important results from numerous papers and books dealing with the theory of Unitary random matrices and functions of random matrices. We give the REFORM method of proving of all results, avoiding the method of moments. We do not try to describe here all known properties of the eigenvalues and eigenvectors for all classes of random matrices. However, our aim is rather to present the theory of stochastic canonical equations, and to give rigorous proofs of the procedures used to deduce these equations on the base of the author's General Statistical Analysis. We consider special classes of analytic functions of random matrices. The description problem for normalized spectral functions of some analytic functions of random matrices is discussed in detail. Specifically, we present here the new theory: LIFE, which is the abbreviation for Limit Independence of Functions of Ensembles. (author)