2025/03/10 by Boris Hanin, Hanin, Boris, T. Jiang +1 · 1 citation
Mathematics · Computer Science · #Random Matrices and Applications #Mathematical Dynamics and Fractals #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.2503.07872
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.