Normal random matrix ensemble as a growth problem
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. Algebrogeometrical 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 algebrogeometrical 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.
 Publication:

Nuclear Physics B
 Pub Date:
 January 2005
 DOI:
 10.1016/j.nuclphysb.2004.10.006
 arXiv:
 arXiv:hepth/0401165
 Bibcode:
 2005NuPhB.704..407T
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter  Mesoscopic Systems and Quantum Hall Effect;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 44 pages, 14 figures