会议地点：中国科学技术大学东区管理科研楼 1418

主办单位：中国科学技术大学数学科学学院

联系人：刘党政，张禄 zl123456@mail.ustc.edu.cn，谷艳东 gd27@mail.ustc.edu.cn

报告题目与摘要

摘要：

In this talk we will focus on the non-asymptotic properties for the spectral norm of structured random matrices with α-subexponential entries. In particular, let X = (Xij ) be an n × n symmetric structured random matrix and denote by∥X∥ the spectral norm of X. Based on an improved ε-net argument, we first establish a deviation inequality for ∥X∥ in the case that Xij is subexponential. Then, via a chaining argument, we obtain an optimal bound for E∥X∥ p in the α-exponential cases with 1 ≤ α ≤ 2. Finally, we study the bound for E∥BX′∥. Here, B is an m×N fixed random matrix, and X′ is an N ×n structured random matrix with independent centered subexponential entries. For this model, we show an upper bound for E∥BX′∥ p , which is in turn used in the study of the smallest singular value of the subexponential random matrix. This talk is mainly based on recent joint works with Dai and Wang (戴国政和王汉超).

摘要：

For finite-rank additive perturbations of inhomogeneous Hermitian random matrices with symmetric sub-Gaussian entries, under the very general assumption that the variance profile is a symmetric stochastic matrix, we establish sharp law of large numbers and fluctuations for spectral outliers that occur as the perturbation strength exceeds a threshold. These limit theorems might depend on both geometric structure and sparsity of the variance profile. This talk is based on joint work with Dang-Zheng Liu and Guangyi Zou.

摘要：

The study of conditioned limit theorems for random walks with independent and identically distributed jumps on the real line has been initiated by Spitzer and Feller, and have attracted the interest of many authors in recent decades. The case of sums of dependent random variables is considerably less explored, mainly due to two challenges. The first one arises from the inapplicability of Wiener-Hopffactorization in this context. The second one is associated with the techniques of reversibility of the random walk, and the issue for dependent random variables is that, in general, the reversed random walk does not exhibit the same dependence structure as the direct random walk. In this talk, we present recent progress on conditioned local limit theorems for products of random matrices, and give potential applications of our results to the study of branching random walks on linear groups and multidimensional Mandelbrot cascades.

摘要：

This talk contains two parts. Part I. We give a new proof of a theorem of Dubrovin, establishing a relationship between the GUE (Gaussian Unitary Ensemble) partition function and the partition function of Gromov–Witten invariants of the complex projective line. Part II. We show that the GUE partition function is equal to part of the topological partition function of the non-linear Schrödinger Frobenius manifold. Connections with Witten’s topological quantum gravity will also be discussed.