### Abstract

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_{n}(x)dx = e^{-nV(x)}dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in [8].

Original language | English (US) |
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Pages (from-to) | 1335-1425 |

Number of pages | 91 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 52 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1999 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*52*(11), 1335-1425. https://doi.org/10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1