Normal random matrix ensemble as a growth problem

R. Teodorescu; E. Bettelheim; O. Agam; A. Zabrodin; P. Wiegmann

doi:10.1016/j.nuclphysb.2004.10.006

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arXiv · hep-th/0401165

Normal random matrix ensemble as a growth problem

R. Teodorescu·E. Bettelheim·O. Agam·A. Zabrodin·P. Wiegmann

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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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R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin, P. Wiegmann. 2004-06-20. Normal random matrix ensemble as a growth problem. https://doi.org/10.1016/j.nuclphysb.2004.10.006

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