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arXiv · 2607.04047

Top Singular Value in Sum-Products of Random Matrices

Kevin Han Huang·Boris Hanin
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Abstract

We study the top singular value for a sum of $m$ independent $n \times n$ random matrices, each of which is a product of $N$ i.i.d. $n\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\rightarrow \infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\beta=\sqrt{2(N-1)/(n\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise.

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Kevin Han Huang, Boris Hanin. 2026-07-04. Top Singular Value in Sum-Products of Random Matrices. https://arxiv.org/abs/2607.04047

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