## 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.

Original language | American English |
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Pages (from-to) | 407-444 |

Number of pages | 38 |

Journal | Nuclear Physics B |

Volume | 704 |

Issue number | 3 |

DOIs | |

State | Published - 10 Jan 2005 |

### Bibliographical note

Funding Information:We are indebted to A. Kapaev, V. Kazakov, I. Krichever, I. Kostov, A. Marshakov and M. Mineev-Weinstein for useful discussions, interest in the subject and help. P.W. and R.T. were supported by the NSF MRSEC Program under DMR-0213745, NSF DMR-0220198 and by the Humboldt foundation. A.Z. and P.W. acknowledge support by the LDRD project 20020006ER “Unstable Fluid/Fluid Interfaces” at Los Alamos National Laboratory and M. Mineev-Weinstein for the hospitality in Los Alamos. A.Z. was also supported in part by RFBR grant 03-02-17373 and by the grant for support of scientific schools NSh-1999.2003.2. P.W. is grateful to K.B. Efetov for the hospitality in Ruhr-Universität Bochum and to A. Cappelli for the hospitality in the University of Florence, where this work was completed. We are grateful to Harry Swinney for permitting us to use Fig. 5 from [16] .

## Keywords

- Integrable systems
- Random matrix theory
- Stochastic growth