2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101986The integrable structure of Ginibre's Orthogonal Ensemble of random matrices is looked at through the prism of the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric Gaussian random matrix. The exact solution for the probability function "p_{n,k}" is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.Published version (4.5 pages, 1 table; title changed, corrected typos)Mathematical PhysicsDisordered Systems and Neural NetworksHigh Energy Physics - TheoryExactly Solvable and Integrable Systemstext