Infinite Products of Large non-Hermitian Random Matrices
发布时间：2019年10月31日
浏览次数：6609
发布者: He Liu
主讲人： Dang-Zheng Liu (University of Science and Technology of China)
活动时间： 从 2019-11-03 15:00 到 16:00
场地： 北京国际数学研究中心，镜春园78号院（怀新园）77201室
Products of M i.i.d. non-Hermitian random matrices of size N × N relate to classical limit theorems in probability theory (N = 1 and large M), to Lyapunov exponents in dynamical systems (finite N and large M), and to local universality in random matrix theory (finite M and large N). The two different limits of M and N for singular values correspond respectively to Gaussian and random matrix theory universality, however, what happens if both M and N go to infinity? This problem, proposed by Akemann, Burda, Kieburg and Deift, lies at the heart of the understanding of both kinds of universal limits. In the cases of complex Gaussian or truncated unitary matrices, we prove that the local eigenvalue statistics undergoes a transition as the relative ratio of M and N changes from 0 to ∞: Ginibre statistics when M/N → 0, Gaussian fluctuation when M/N → ∞, and new critical phenomena when M/N →γ ∈ (0, ∞). Joint work with Yanhui Wang, Henan University.
