Introduction to Random-Matrix Theory

Random-matrix theory gained attention during the 1950s due to work by Eugene Wigner in mathematical physics. Specifically, Wigner wished to describe the general properties of the energy levels (or of their spacings) of highly excited states of heavy nuclei as measured in nuclear reactions (Wigner, 1957). Such a complex nuclear system is represented by an Hermitian operator H (called the Hamiltonian) living in an infinite-dimensional Hilbert space governed by physical laws. Unfortunately, in any specific case, H is unknown. Moreover, even if it were known, it would be much too complicated to write down and, even if we could write it down, no computer would be able to solve its eigenequation Hv = λv (the so-called Schrödinger equation of the physical system), where λ and v are an eigenvalue-eigenvector pair corresponding to H.