We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent i.i.d. random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$.
We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent i.i.d. random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances.
The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed sub-Gaussian entries. This lemma is of independent interest.
"Random matrix products: Universality and least singular values." Ann. Probab. 48 (3) 1372 - 1410, May 2020. https://doi.org/10.1214/19-AOP1396