Global Universality of Singular Values in Products of Many Large Random Matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929752542019584 |
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| author | Hanin, Boris Jiang, Tianze |
| author_facet | Hanin, Boris Jiang, Tianze |
| contents | We study the singular values (and Lyapunov exponents) for products of $N$ independent $n\times n$ random matrices with i.i.d. entries. Such matrix products have been extensively analyzed using free probability, which applies when $n\to \infty$ at fixed $N$, and the multiplicative ergodic theorem, which holds when $N\to \infty$ while $n$ remains fixed. The regime when $N,n\to \infty$ 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 $n, N\to \infty$ 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 $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07872 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Universality of Singular Values in Products of Many Large Random Matrices Hanin, Boris Jiang, Tianze Probability Mathematical Physics We study the singular values (and Lyapunov exponents) for products of $N$ independent $n\times n$ random matrices with i.i.d. entries. Such matrix products have been extensively analyzed using free probability, which applies when $n\to \infty$ at fixed $N$, and the multiplicative ergodic theorem, which holds when $N\to \infty$ while $n$ remains fixed. The regime when $N,n\to \infty$ 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 $n, N\to \infty$ 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 $N$. |
| title | Global Universality of Singular Values in Products of Many Large Random Matrices |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2503.07872 |