## Abstract

Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form W_{n}^{2 n} (x) d x in the region where universality is desired. W_{n} does not need to be analytic, nor possess more than one derivative-and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form W^{2 n} (x) d x, for example, when W = e^{- Q} where Q is convex and Q^{′} satisfies a Lipschitz condition of some positive order. We also deduce universality for a class of fixed exponential weights on a real interval.

Original language | English |
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Pages (from-to) | 743-779 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 219 |

Issue number | 3 |

DOIs | |

State | Published - 20 Oct 2008 |

### Bibliographical note

Funding Information:Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353. Corresponding author. E-mail addresses: elile@openu.ac.il (E. Levin), lubinsky@math.gatech.edu (D.S. Lubinsky).

## Keywords

- Orthogonal polynomials potential theory with external fields
- Random matrices
- Universality limits