Weaknesses: - The presentation can be largely improved: some of the sentences do not help the reader and currently the draft is not easy to read. For instance, the transition to the second paragraph of the introduction is quite abrupt: while the first paragraph talks about keypoints and images, the keypoints disappear when talking about the permutation synchronization. Moreover, the sigma is not properly defined. Similarly, Section 4.1 is hard to read: while the overall idea is clear, the reader gets trapped in too many details, such as those in lines 196-199. Similar comments hold for Section 4.2: why presenting two techniques when you recommend using only one? While I understand the desire of the authors to be comprehensive, mentioning too many details may compromise clarity.

Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the algorithms based on GCL. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries of fixed size. Part of the theory covers the case where there is significant dependence between the blocks. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.