Universal relation between Green functions in random matrix theory Original Research Article

We prove that in random matrix theory there exists a universal relation between the onepoint Green function G and the connected two-point Green function Gc given by N2Gc(z, w) = (∂2/∂z ∂w) log[(G(z) − G(w))/(z − w) + irrelevant factorized terms]. This relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though G is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered some time ago, which states that a2Gc (az, aw) is independent of the probability distribution, where a denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively by Brézin, Hikami, and Zee.