Law of addition in random matrix theory

We discuss the problem of adding random matrices, which enable us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of ``gluon connectedness," 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 Br\'ezin, Hikami, and Zee. The method used here may be applied to a broad class of problems involving random matrices.