Collections of large matrices, chosen at random from some ensemble. Random-matrix theory is a branch of mathematics which emerged from the study of complex physical problems, for which a statistical analysis is often more enlightening than a hopeless attempt to control every degree of freedom, or every detail of the dynamics. Although the connections to various parts of mathematics are very rich, the relevance of this approach to physics is also significant.
Random matrices were introduced by Eugene Wigner in nuclear physics in 1950. In quantum mechanics the discrete energy levels of a system of particles, bound together, are given by the eigenvalues of a hamiltonian operator, which embodies the interactions between the constituents. This leads to the Schrödinger equation which, in most cases of interest in the physics of nuclei, cannot be solved exactly, even with the most advanced computers. For a complex nucleus, instead of finding the location of the nuclear energy levels through untrustworthy approximate solutions, Wigner proposed to study the statistics of eigenvalues of large matrices, drawn at random from some ensemble. The only constraint is to choose an ensemble which respects the symmetries that are present in the forces between the nucleons in the original problem, and to select a sequence of levels corresponding to the quantum numbers that are conserved as a consequence of these symmetries, such as angular momentum and parity. The statistical theory does not attempt to predict the detailed sequence of energy levels of a given nucleus, but only the general properties of those sequences and, for instance, the presence of hidden symmetries. In many cases this is more important than knowing the exact location of a particular energy level. This program became the starting point of a new field, which is now widely used in mathematics and physics for the understanding of quantum chaos, disordered systems, fluctuations in mesoscopic systems, random surfaces, zeros of analytic functions, and so forth. See Conservation laws (physics), Eigenvalue (quantum mechanics), Quantum mechanics
The mathematical theory underlying the properties of random matrices overlaps with several active fields of contemporary mathematics, such as the asymptotic behavior of orthogonal polynomials at large-order, integrable hierachies, tau functions, semiclassical expansions, combinatorics, and group theory; and it is the subject of active research and collaboration between physics and mathematics.