Global Universality of Singular Values in Products of Many Large Random Matrices

We study the singular values (and Lyapunov exponents) for products of N independent n n random matrices with i. i. d. entries. Such matrix products have been extensively analyzed using free probability, which applies when n at fixed N, and the multiplicative ergodic theorem, which holds when N while n remains fixed. The regime when N, n simultaneously is considerably less well understood, and our work is the first to prove universality for the global distribution of singular values in this setting. Our main result gives non-asymptotic upper bounds on the Kolmogorov-Smirnoff distance between the empirical measure of (normalized) squared singular values and the uniform measure on 0, 1 that go to zero when n, N at any relative rate. We assume only that the distribution of matrix entries has zero mean, unit variance, bounded fourth moment, and a bounded density. Our proofs rely on two key ingredients. The first is a novel small-ball estimate on singular vectors of random matrices from which we deduce a non-asymptotic variant of the multiplicative ergodic theorem that holds for growing matrix size n. The second is a martingale concentration argument, which shows that while Lyapunov exponents at large N are not universal at fixed matrix size, their empirical distribution becomes universal as soon as the matrix size grows with N.