MATRIX MODELS AND GROWTH PROCESSES: FROM VISCOUS FLOWS TO THE QUANTUM HALL EFFECT

摘要

We review the recent developments in the theory of normal, normal self-dual and general complex random matrices. The distribution and correlations of the eigenvalues at large scales are investigated in the large N limit. The 1/N expansion of the free energy is also discussed. Our basic tool is a specific Ward identity for correlation functions (the loop equation), which follows from invari-ance of the partition function under reparametrizations of the complex eigenvalues plane. The method for handling the loop equation requires the technique of boundary value problems in two dimensions and elements of the potential theory. As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g.. to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.