2018/09/30 by Dang-Zheng Liu, Dong Wang, Liu, Dang-Zheng +3 · 4 citations
Mathematics · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1810.00433
Products of M i.i.d. random matrices of size N × N are related 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 universality in random matrix theory (finite M and large N). Under the two different limits of M → ∞ and N → ∞, the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both M and N go to infinity. This problem, proposed by Akemann, Burda, Kieburg \citeAkemann-Burda-Kieburg14 and Deift \citeDeift17, lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of M and N changes from 0 to ∞: sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when M/N → 0, Gaussian fluctuation when M/N → ∞, and new critical phenomena when M/N → γ∈ (0,∞). Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy-Widom distributions.