Determinism plus Chance in Random Matrix Theory

We study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of " gluon connected-ness, " we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Brézin, Hikami, and Zee. The method used here may be applied to a broad class of problems involving random matrices. Some four decades ago, Wigner 1 proposed studying the distribution of energy levels of a random Hamiltonian given by H = ϕ (1) where ϕ is an N by N hermitean matrix taken from the distribution P (ϕ) = 1 Z e −N trV (ϕ). (2) with Z fixed by dϕP (ϕ) = 1. This problem has been studied intensively by Dyson, Mehta, and others over the years. 2,3,4 Two years ago, Brézin and Zee discovered that, remarkably, while the density of eigenvalues depends 5 on V , the correlation between the density of eigenvalues, when suitably scaled, is independent 6 of V. This universality has been clarified and extended by other authors, 7,8,9 studied numerically, 10 and furthermore, shown to hold even when the distribution (2) is generalized to a much broader class of distributions. 11 We expect that the discussion to be given below will hold also for this broader class of distributions, but for the sake of simplicity we will not work this through here.