Infinite Products of Large non-Hermitian Random Matrices
Time: 2019-10-31
Published By: He Liu
Speaker(s): Dang-Zheng Liu (University of Science and Technology of China)
Time: 15:00-16:00 November 3, 2019
Venue: Room 77201, Jingchunyuan 78, BICMR
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.
