Dang-Zheng Liu

 

Affiliations:
School of Mathematical Sciences
University of Science and Technology of China (USTC)

 

Office:
Room 1520
Management and Research Building
East Campus, University of Science and Technology of China
Hefei, Anhui, 230026 China
Email: dzliu at ustc dot edu dot cn

 

If you are interested in Random Matrix Theory or its applications to Representation Theory, Number Theory, Statistics and Finance, please feel free to contact me.

Biography

Dang-Zheng Liu received his Ph.D. degree in Mathematics from Peking University, Beijing, under the supervision of Prof. Zheng-Dong Wang. Before that, he got his Bachelor degree in Mathematics from Shannxi Normal University in 2005. From October 2010 to August 2012, he was a postdoctor in University of Talca, Chile. He joined the School of Mathematical Sciences first as a tenure-track associate professor in September 2012 and then as an associate professor in February 2016.

Research

His research interests are Random Matrix Theory and related applications.

随机矩阵=矩阵+概率论

矩阵和随机现象的无处不在预示蕴含了随机矩阵的处处存在。概率论、多元统计、表示论、计数组合、可积系统、弦论、统计物理等研究领域的思想方法融汇在矩阵里，而随机矩阵也会神秘地出现在看似毫不相关的领域中，尤其体现在它和黎曼zeta函数密切的关联上。而且它莫名其妙地出席的场合还在不断地增加。 随机矩阵主要研究矩阵维数趋于无穷时特征值和特征向量的渐近性质，尤其是那些展现出某些特定模式(Pattern)的普适性质(Universality)，这有点类似概率论里研究各种极限定理的风格。然而，随机矩阵的普适性质有着更丰富的内容，也需要更广泛的技巧方法。 随机矩阵是当前非常活跃的研究领域，既有丰富多样的重要问题，也给许多数学技术方法提供了演示的舞台。 可以想象随着大维数据和复杂结构的兴起，随机矩阵的重要性和应用会有所体现。

正在研究的问题：

A. β-系综的关联函数和特征多项式。 从统计物理的观点看，Dyson指标β从β=1, 2, 4的正交、酉、辛随机矩阵系综过渡到任意正数的 β-系综是很自然的一步，而且典型的β-系综可实现为三对角随机矩阵模型(Dumitriu-Edelman实现)，进而和随机算子发生联系。 B. 随机矩阵乘积的奇异特征值和复特征值。从单个随机矩阵到多个随机矩阵的研究是个很困难的问题，而随机矩阵乘积可看做一个中间状态。关注的一个问题是大维随机矩阵的无穷乘积Lyapunov指数的统计性质。 C. 随机块状与带状矩阵。带状矩阵可以看做从随机Schrodinger算子到随机矩阵的插值，当带宽在某个临界值附近时会出现Anderson相变，这是一个重要但非常困难的问题。

对随机矩阵理论好奇者可以读读维基词条 Random Matrix, 或者一篇很优秀的高级科普“At the Far Ends of a New Universal Law” A new universal law

 

Teaching

  • 09/2013~01/2014: Probability Theory
  • 02/2014~06/2014: Probability Theory
  • 09/2014~01/2015: Probability Theory
  • 02/2015~06/2015: Probability Theory
  • 09/2015~01/2016: Probability Theory
  • 02/2016~06/2016: Probability Theory, Stochastic Analysis (With Elton P Hsu, Ran Wang)
  • 09/2017~01/2018: Advanced Probability Theory try
  • 02/2018~06/2018: Stochastic Process
 

Selected Publications

 

 

Some universal properties for restricted trace Gaussian orthogonal, unitary and symplectic ensembles

 

Liu, D.-Z., Zhou, D.-S.

 

J. Stat. Phys. 140: 268--288, 2010. paper

 

 

 

Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

 

Liu, D.-Z., Wang, Z.-D.

 

J. Theoret. Probab.24 (4), 988--1001, 2011. paper

 

 

 

Local statistical properties of Schmidt eigenvalues of bipartite entanglement for a random pure state

 

Liu, D.-Z., Zhou, D.-S.

 

Int. Math. Res. Notices 2011 (4):725--766, 2011. paper

 

 

 

On Explicit Probability Densities Associated with Fuss-Catalan Numbers

 

Liu, D.-Z., Song, C., Wang, Z.-D.

 

Proc. Amer. Math. Soc. 139, 3735--3738, 2011. paper

 

 

 

Fluctuations of Eigenvalues for Random Toeplitz and Related Matrices

 

Liu, D.-Z., Sun, X., Wang, Z.-D.

 

Electron. J. Probab. 17, no. 95, 1--22, 2012. paper

 

 

 

Asymptotics for products of characteristic polynomials in classical β-ensembles

 

Desrosiers, P., Liu, D.-Z.

 

Constructive Approximation 39, no. 2, 273-322, 2014.paper

 

 

 

Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian β-Ensemble with External Source

 

Desrosiers, P., Liu, D.-Z.

 

Int. Math. Res. Notices, Vol. 2015, No. 12, 3751–3781, 2015. paper

 

 

 

Raney distributions and random matrix theory

 

Forrester, P.J., Liu, D.-Z.

 

J. Stat. Phys. 158, no.5, 1051-1082, 2015. paper

 

 

 

Selberg integrals, super-hypergeometric functions and applications to β-ensembles of random matrices

 

Desrosiers, P., Liu, D.-Z.

 

Random Matrices Theory Appl. 4, no.2, 1550007, 59pp, 2015.

 

 

 

Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition

 

Forrester, P.J., Liu, D.-Z.

 

Communications in Mathematical Physics, vol.344, no.1, 333-368, 2016.paper

 

 

 

Universality for products of random matrices I: Ginibre and truncated unitary cases

 

Liu, D.-Z., Wang, Y.

 

Int. Math. Res. Notices, vol. 2016, no.7, 3473-3524, 2016. paper

 

 

 

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

 

Liu, D.-Z., Wang, D., Zhang, L.

 

Annales de l'Institut Henri Poincare (B) Probabilités et Statistiques, vol.52, no.4, 1734-1762, 2016. paper

 

 

 

Singular values for products of two coupled random matrices: hard edge phase transition

 

Liu, D.-Z.

 

Constructive Approximation, 42 pages, online doi:10.1007/s00365-017-9389-z. paper

 

 

 

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

 

Akemann, G, Checinski, T, Liu, D.-Z., Strahov, E.

 

To appear in AIHP Probabilite et Statistique paper

 

 

 

Matrix product ensembles of Hermite type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral

 

Forrester, P.J., Ipsen, J. R., Liu, D.-Z.

 

Ann. Henri Poincare, 42 pages paper

 

 Students

  • Tiantian Chen , 9/2016 - ;
  • Shuang Li , 9/2016 - ;
  • Yu Xiang , 9/2016 - ;
  • Lu Zhang , 9/2017 - ;

Workshop

Workshop on Stochastic Analysis and Random Matrices, 8-10 December, 2017. Place: School of Mathematical Sciences, USTC, Hefei. Organizers: Xiang-Dong Li (Institute of Applied Mathematics, AMSS, CAS), Dang-Zheng Liu (School of Mathematical Sciences, USTC) SARM

Mini-Courses

Random matrix theory and log-correlated Gaussian fields, NJ Simm, 26-26 March, 2018. Place: School of Mathematical Sciences, USTC, Hefei. SARM

Copyright © Juyong Zhang