Diagrammatic Perturbation Expansion for Ensembles of Random Matrices

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Abstract
A method for obtaining a perturbation expansion of various eigenvalue distributions corresponding to a certain class of perturbed ensembles of random matrices is given. The terms in the expansion can be written down immediately as diagrams analogous to those used in other kinds of perturbation theory. Further, part of the expansion can be summed explicitly, and the result of the summation read off the diagrams. In addition, a new perturbing ensemble is introduced. It has the advantage that the number of matrix elements which are perturbed simultaneously and the size of the perturbation can be varied independently. The expansion given is an expansion in the number of perturbed matrix elements rather than the usual expansion in the size of the elements. Finally, the conditions for convergence of the expansion are discussed.