Normal random matrix ensemble as a growth problem |
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| Institution: | 1. James Frank Institute, University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA;2. Racah Institute of Physics, Hebrew University, Givat Ram, Jerusalem 91904, Israel;3. Institute of Biochemical Physics, Kosygina str. 4, 117334 Moscow, Russia;4. James Frank Institute, Enrico Fermi Institute, University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA;1. Lebedev Physics Institute, Moscow 119991, Russia;2. ITEP, Moscow 117218, Russia;3. Institute for Information Transmission Problems, Moscow 127994, Russia;4. National Research Nuclear University MEPhI, Moscow 115409, Russia;5. Institute of Nuclear Research, Moscow 117312, Russia;1. Department of Mathematics, King''s College London, Strand, London WC2R 2LS, UK;2. Physik Department and Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany;1. Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India;2. Physics Department, McGill University, 3600 University St, Montréal, QC H3A 2T8, Canada;1. Lebedev Physics Institute, Moscow 119991, Russia;2. ITEP, Moscow 117218, Russia;3. Institute for Information Transmission Problems, Moscow 127994, Russia;4. MIPT, Dolgoprudny 141701, Russia;1. Dipartimento di Matematica e Fisica, Università Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy;2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, I-00133 Roma, Italy |
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| Abstract: | In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples. |
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