The application of random matrix theory to observed data structures (e.g. graphs) has led to general interest in eigenvalue concentration inequalities. Proofs for such inequalities often make the distinct eigenvalue assumption, which leads to the existence of eigengaps, followed by the application of canonical results such as the DavisKahan theorem.
We address the concentration of eigenvalues in the finitedimensional random dot product graph model. More specifically, we prove under mild assumptions that the eigenvalues of the adjacency spectral embedding concentrate around the true eigenvalues even when the true eigenvalues have multiplicity. Our result extends naturally to a more general finitedimensional signalnoise model with upper tail probability bounds.
