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We find the exact per-node capacity scaling in an extended, distributed wireless network where the node locations are random, the channel attenuation between pairs of nodes exhibits independent random fading, and data forwarding is restricted to multihop delivery. We establish a per-node throughput upper bound equal to (Cmax)/(radicn) (bps)/(Hz), and a constructive lower bound equal to (Cmin)/(radicn) (bps)/(Hz), where Cmax and Cmin are constants and n is the expected number of nodes in the network. The results apply with probability approaching unity as n becomes large, under independent, frequency flat, fading channel conditions where the tail probability exhibits an exponential decay (e.g., any mixture of line-of-sight and Rayleigh, Rice and Nakagami distributions). Our lower bound is mainly motivated by a clever bond-percolation-based protocol construction, introduced by Franceschetti et al., where it was shown that a per-node throughput equal to a constant times 1/(radicn) (bps)/(Hz) is achievable in networks with random node locations under a deterministic channel gain modeling path-loss and absorption. We extend the result to more realistic channel gain models in which the channel gains are modeled as random due to multipath effects. The protocol systematically leverages multiuser diversity to overcome the added uncertainty. The key to the upper bound is to establish an upper bound for the total network transport capacity. This upper bound proves that, relative to the lower bound construction, any attempt to leverage multiuser diversity further will result in no more than a constant factor of throughput gain.