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We consider dense wireless sensor networks deployed to observe arbitrary random fields. The requirement is to reconstruct an estimate of the random field at a certain collector node. This creates a many-to-one data gathering wireless channel. In this note, we first characterize the transport capacity of many-to-one dense wireless networks subject to a constraint on the total average power. In particular, we show that the transport capacity scales as Theta(log(N)) when the number of sensors N grows to infinity and the total average power remains fixed. We then use this result along with some information-theoretic tools to derive sufficient and necessary conditions that characterize the set of observable random fields by dense sensor networks. In particular, for random fields that can be modeled as discrete random sequences, we derive a certain form of source/channel coding separation theorem. We further show that one can achieve any desired nonzero mean-square estimation error for continuous, Gaussian, and spatially bandlimited fields through a scheme composed of single-dimensional quantization, distributed Slepian-Wolf source coding, and the proposed antenna sharing strategy. Based on our results, we revisit earlier conclusions about the feasibility of data gathering applications using dense sensor networks.