浙江大学随机矩阵理论与应用研讨会

  浙江大学随机矩阵理论与应用研讨会

2024年9月28日

浙江大学数学科学学院海纳苑2号楼206

报告题目与摘要

报告一：张仑，复旦大学

时间：9:30-10:30

题目：The multiplicative constant in asymptotics of higher-order analogues of the Tracy-Widom distribution

摘要：In this talk, we are concerned with higher-order analogues of the Tracy-Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by functions related to the even member of the Painleve I hierarchy P_l^2k $, and are regarded as higher-order analogues of the Airy kernel. We present a novel approach to establish the multiplicative constant in asymptotics of the distribution, resolving an open problem in the work of Clayes, Its and Krasovsky. An important new feature of the expression is the involvement of an integral of the Hamiltonian associated with a special, real, pole free solution for P_l^2k. In addition, we show that the total integral of the Hamiltonian vanishes for all $k$, and establish a transition from the higher-order Tracy-Widom distribution to the classical one. Our approach can also be adapted to calculate similar critical constants in other problems arising from mathematical physics. Joint work with Dan Dai, Wen-Gao Long, Shuai-Xia Xu and Lu-Ming Yao

报告二：王东，中国科学院大学

时间：10:30-11:30

题目：Local statistics of Muttalib-Borodin ensemble in the hard to soft transitive regime

摘要：Muttalib-Borodin ensemble is a typical biorthogonal ensemble. It has a hard edge limit that is expressed by Wright's generalized Bessel functions, or by Meijer G functions if the parameter θ is an integer. This hard edge limit does not occur in orthogonal polynomial ensembles. In this talk we consider the transition from the hard edge limit to the soft edge limit of the Muttalib-Borodin ensemble with an integer parameter θ , and find an interpolation from the Meijer G correlation kernel to the Airy correlation kernel. This is joint work with Shui-Xia Xu.

 报告三：姚东，江苏师范大学

时间：14:00-15:00

题目：Small gap problem of point processes

摘要：In the talk, I will give a brief review of known results on the extreme gap problems (smallest and largest gaps of the eigenvalues) of various random matrix ensembles. Then I will present our recent series of work on smallest gaps of several random point processes with Gaussian structures

报告四：刘党政，中国科技大学

时间：15:00-16:00

题目：随机带状矩阵的新进展

摘要：诺贝尔物理奖获得者Wigner和 Anderson 在上世纪50年代独立地将随机性引入到量子物理的哈密尔顿量中，Wigner矩阵和Anderson模型从此诞生。 作为具有稠密结构的平均场模型和稀疏结构的非平均场模型，两者看似非常不同又各自发展。然而其背后却隐藏神秘的相似性，随机带状矩阵则是联通两者的一座桥梁。本报告介绍随机带状矩阵一些最新进展与思考。

报告五：鲍志刚，香港大学

时间：16:20-17:20

题目：A leave-one-out approach to approximate message passing

摘要：Approximate message passing (AMP) has emerged both as a popular class of iterative algorithms and as a powerful analytic tool in a wide range of statistical estimation problems and statistical physics models. A well established line of AMP theory proves Gaussian approximations for the empirical distributions of the AMP iterate in the high dimensional limit, under the GOE random matrix model and its variants.  In this talk, we will introduce a non-asymptotic, leave-one-out representation for the AMP iterate that holds under a broad class of Gaussian random matrix models with general variance profiles. In contrast to the typical AMP theory that describes the empirical distributions of the AMP iterate via a low dimensional state evolution, our leave-one-out representation yields an intrinsically high dimensional state evolution formula which provides non-asymptotic characterizations for the possibly heterogeneous, entry-wise behavior of the AMP iterate under the prescribed random matrix models. Our leave-one-out method of proof differs significantly from the widely adopted conditioning approach for rotational invariant ensembles, and relies instead on an inductive method that utilizes almost solely integration-by-parts and concentration techniques. This talk is based on a joint work with Qiyang Han and Xiaocong Xu