  We consider a dense $K$ user Gaussian interference network formed by paired
transmitters and receivers placed independently at random in a fixed spatial
region. Under natural conditions on the node position distributions and signal
attenuation, we prove convergence in probability of the average per-user
capacity $\csum/K$ to $\half \ep \log(1 + 2 \SNR)$. The achievability result
follows directly from results based on an interference alignment scheme
presented in recent work of Nazer et al. Our main contribution comes through an
upper bound, motivated by ideas of `bottleneck capacity' developed in recent
work of Jafar. By controlling the physical location of transmitter--receiver
pairs, we can match a large proportion of these pairs to form so-called
$\epsilon$-bottleneck links, with consequent control of the sum capacity.
