In this paper, a lower bound on the capacity of wireless ad hoc erasure networks is derived in closed form in the case where n nodes are uniformly and independently distributed in the unit area square. It holds almost surely and is asymptotically tight. Nodes are assumed to have fixed transmit power; hence, two nodes should be within a specified distance r_n to overcome noise. With interference determining outages, each transmitter-receiver pair is modeled as an erasure channel with a broadcast constraint, i.e., each node can transmit only one signal across all its outgoing links. A lower bound of Θ(nr_n) for the network capacity is derived when erasures across distinct links are independent, with constant erasure probabilities. When erasures are correlated, the lower bound Θ(1/(r_n)) is proved. If the broadcast constraint is relaxed, the gain is a function of r_n and the link erasure probabilities, and is at most a constant if the erasure probabilities grow sufficiently large with n. Finally, the case where the erasure probabilities are random variables, for example due to randomness in geometry or channels, is analyzed. In this setting, it is shown somewhat surprisingly that variability in erasure probabilities increases network capacity.