Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
✍ Scribed by P. Deift; T. Kriecherbauer; K. T-R McLaughlin; S. Venakides; X. Zhou
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 475 KB
- Volume
- 52
- Category
- Article
- ISSN
- 0010-3640
No coin nor oath required. For personal study only.
✦ Synopsis
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 dµ V for V as analyzed in [8].