Non-Hermitian random matrix theory: Summation of planar diagrams, the 'single-ring' theorem and the disc-annulus phase transition

I review aspects of work done in collaboration with A Zee and R Scalettar (1997 Nucl. Phys. B 504 579; 1997 Nucl. Phys. B 501 643; 2001 J. Math. Phys. 42 5718) on complex non-Hermitian random matrices. I open by explaining why the bag of tools used regularly in analysing Hermitian random matrices cannot be applied directly to analyse non-Hermitian matrices, and then introduce the method of Hermitization, which solves this problem. Then, for rotationally invariant ensembles, I derive a master equation for the average density of eigenvalues in the complex plane, in the limit of infinitely large matrices. This is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the Hermitized Green function. Remarkably, this resummation can be carried out explicitly for any rotationally invariant ensemble. I prove that in the limit of infinitely large matrices, the shape of the eigenvalue distribution is either a disc or an annulus. This is the celebrated 'single-ring' theorem. Which of these shapes is realized is determined by the parameters (coupling constants) which determine the ensemble. By varying these parameters a phase transition may occur between the two possible shapes. I briefly discuss the universal features of this transition. As the analysis of this problem relies heavily on summation of planar Feynman diagrams, I make a special effort at presenting a pedagogical exposition of the diagrammatic method, which some readers may find useful.