2004/01/31 by R. Teodorescu, Razvan Teodorescu, E. Bettelheim +5 · 2 citations
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Random Matrices and Applications #Stochastic processes and statistical mechanics #cond-mat.mes-hall #hep-th #nlin.SI
paper · pdf · doi:10.1016/j.nuclphysb.2004.10.006
published as Nucl.Phys. B704 (2005) 407-444 · 44 pages, 14 figures; contains the first part of the original file. The second part will be submitted separately
arxiv created 2004/06/20 · openalex publication_date 2004/12/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.