In this paper we consider the MIMO broadcast channel with antenna correlation at the transmitter and receiver. We derive the theoretical sum rate of systems with a large number of antennas for zero-forcing and regularized zero-forcing precoders. Particularly, we apply the results to volume-limited devices where the correlation originates from a dense antenna packing. Throughout this contribution we make extensive use of recent tools from random matrix theory. Simulations confirm the theoretical claims and also indicate that in most scenarios the asymptotic derivations applied to a finite number of users give good approximations of the true ergodic sum rate.