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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 CΣ/K to 1/2E log(1 + 2SNR). 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 ε-bottleneck links, with consequent control of the sum capacity.