  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.
