Vector diffusion maps and random matrices with random blocks

Vector diffusion maps (VDM) 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 VDM, 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 VDM algorithm - whether there is signal or not. 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. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.