Normal random matrix ensemble as a growth problem

Details Normal random matrix ensemble as a growth problem Normal random matrix ensemble as a growth problem

2004-01-23

arxiv.org/abs/hep-th/0401165v2

Nucl.Phys. B704 (2005) 407-444

Physics

High Energy Physics

High Energy Physics - Theory

44 pages, 14 figures; contains the first part of the original file. The second part will be submitted separately

Scientific paper

10.1016/j.nuclphysb.2004.10.006

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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