Local statistics in unitarily invariant random matrices
数学专题报告
报告题目(Title)：Local statistics in unitarily invariant random matrices
报告人(Speaker)：王东（中国科学院大学）
地点(Place)：教九 501（2024/10/25），后主楼 1124（2024/11/1）
时间(Time)：2024 年 10 月 25 日，11 月 1 日 14h - 17h
邀请人(Inviter)：陈昕昕
报告摘要
We consider the local limiting distribution of eigenvalues of unitarily invariant random matrices, like the complex Wishart ensemble. The celebrated Tracy-Widom
distribution is the most important example of the limiting local distributions. We consider both the Tracy- Widom distribution and its generalizations. The main method is the analysis of orthogonal polynomials via Riemann-Hilbert problem.
Lecture plan:
1. From Random matrices to the distribution of the eigenvalues
2. Contour integral formulas for eigenvalues in Wishart ensemble
3. Universality of local limiting distributions
4. Transition between hard edge and soft edge universal distributions
参考文献：
[1] Terence Tao. Topics in random matrix theory, volume 132 of Graduate Studies in
Mathematics. American Mathematical Society, Providence, RI, 2012.
[2] P. A. Deift. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, volume 3 of Courant Lecture Notes in Mathematics. New York University
Courant Institute of Mathematical Sciences, New York, 1999.
[3] Alexander R. Its, Arno B. J. Kuijlaars, and Jörgen Östensson. Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent. Int. Math. Res. Not. IMRN, (9):Art. ID rnn017, 67, 2008.
[4] Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov. Painlevé transcendents, volume 128 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. The RiemannHilbert approach.
