2004/12/19 by A. Zabrodin, Zabrodin, A. · 4 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Theoretical and Computational Physics #cond-mat.mes-hall #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0412219
51 pages, 8 figures, Based on the lectures given at the School "Applications of Random Matrices in Physics", Les Houches, June 2004
arxiv created 2004/12/19 · openalex publication_date 2004/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 invariance 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.