Spectral Theory for Products of Many Large Gaussian Matrices

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Boris Hanin, Princeton

Let $X_{N,n}$ be an iid product of $N$ real Gaussian matrices of size $n \times n$. In this talk, I will explain some recent joint work with G. Paouris (arXiv:2005.08899) about a non-asymptotic analysis of the singular values of $X_{N,n}$. I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of $X_{N,n}$ to the so-called Triangle Law in the limit where $N,n$ tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large $N$ on the distance between the joint distribution of all Lyapunov exponents of $X_{N,n}$ and appropriately normalized independent Gaussians in the near-ergodic regime ($N\gg n$).