Large n limit of Gaussian random matrices with external source, Part II

被引:103
作者
Aptekarev, AI
Bleher, PM
Kuijlaars, ABJ
机构
[1] Russian Acad Sci, MV Keldysh Appl Math Inst, Moscow 125047, Russia
[2] Indiana Univ Purdue Univ, Dept Math Sci, Indianapolis, IN 46202 USA
[3] Katholieke Univ Leuven, Dept Math, B-3001 Heverlee, Belgium
基金
美国国家科学基金会;
关键词
D O I
10.1007/s00220-005-1367-9
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We continue the study of the Hermitian random matrix ensemble with external source 1/Zn e(-nTr(1/2M2 - AM))dM where A has two distinct eigenvalues +/- a of equal multiplicity. This model exhibits a phase transition for the value a = 1, since the eigenvalues of M accumulate on two intervals for a > 1, and on one interval for 0 < a < 1. The case a > 1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0 < a < 1. As in Part I we apply the Deift/Zhou steepest descent analysis to a 3 x 3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann- Hilbertproblems.
引用
收藏
页码:367 / 389
页数:23
相关论文
共 37 条
[1]   Multiple orthogonal polynomials for classical weights [J].
Aptekarev, AI ;
Branquinho, A ;
Van Assche, W .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2003, 355 (10) :3887-3914
[2]  
Baik J, 2000, COMMUN PUR APPL MATH, V53, P1385, DOI 10.1002/1097-0312(200011)53:11<1385::AID-CPA3>3.0.CO
[3]  
2-T
[4]   Large n limit of Gaussian random matrices with external source, part I [J].
Bleher, P ;
Kuijlaars, ABJ .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2004, 252 (1-3) :43-76
[5]   Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model [J].
Bleher, P ;
Its, A .
ANNALS OF MATHEMATICS, 1999, 150 (01) :185-266
[6]   Double scaling limit in the random matrix model: The Riemann-Hilbert approach [J].
Bleher, P ;
Its, A .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2003, 56 (04) :433-516
[7]  
Bleher PM, 2004, INT MATH RES NOTICES, V2004, P109
[8]  
BLEHER PM, 2004, IN PRESS ANN I FOURI
[9]  
BLEHER PM, UNPUB LARGE N LIMI 3
[10]   Level spacing of random matrices in an external source [J].
Brézin, E ;
Hikami, S .
PHYSICAL REVIEW E, 1998, 58 (06) :7176-7185