UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY
UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY
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DOI：
10.1002/(sici)1097-0312(199911)52:11
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发表时间：
1999-11
影响因子：
3
通讯作者：
P. Deift;T. Kriecherbauer;K. Mclaughlin;S. Venakides;Xin Zhou
中科院分区：
文献类型：
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作者：
P. Deift;T. Kriecherbauer;K. Mclaughlin;S. Venakides;Xin Zhou
我们考虑正交多项式关于线上的变化指数权重wn（x）dx = e−nV（x）dx的渐近性为n ∞。假定势V是真实的解析的，在无穷远处有足够的增长。的原则结果关注Plancherel-Rotach型渐近正交多项式的轴。利用这些渐近性，我们证明了随机矩阵模型理论中各种统计量的普适性，其中一些最近在[31]和[4]中已经考虑过。另一个应用涉及正交多项式的递归系数和前导系数的渐近性（参见[4]）。正交多项式问题被公式化为下面的Riemann-Hilbert问题[19，20]。Riemann-Hilbert问题依次使用在[12]中引入并在[11，13]中进一步发展的最速下降法进行分析。在我们的方法中，V的平衡测度dμV起着关键作用，如[8]中所分析的。© 1999 John Wiley & Sons，Inc.
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(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]. © 1999 John Wiley & Sons, Inc.