Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).

Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt n}$. I wonder if there are any results giving estimates for the spectral norm of the matrix $\frac{XY}{\sqrt n}$ around its mean with high probability as $n \to \infty$.

As a side question, I also wonder if we could have estimates for the analogous problem for sub-Gaussian matrices (entries are i.i.d sub-Gaussian). Again we require the product $XY$ be normalized by the standard variance of entries.

The main difficulty for this problem is that the entries in the matrix $XY$ are no longer i.i.d.