Top Singular Value in Sum-Products of Random Matrices

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.