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New large deviations results that characterize the asymptotic informationrates for general $d$-dimensional ($d$-D) stationary Gaussian fields areobtained. By applying the general results to sensor nodes on a two-dimensional(2-D) lattice, the asymptotic behavior of ad hoc sensor networks deployed overcorrelated random fields for statistical inference is investigated. Under a 2-Dhidden Gauss-Markov random field model with symmetric first order conditionalautoregression and the assumption of no in-network data fusion, the behavior ofthe total obtainable information [nats] and energy efficiency [nats/J] definedas the ratio of total gathered information to the required energy is obtainedas the coverage area, node density and energy vary. When the sensor nodedensity is fixed, the energy efficiency decreases to zero with rate$\Theta({area}^{-1/2})$ and the per-node information under fixed per-nodeenergy also diminishes to zero with rate $O(N_t^{-1/3})$ as the number $N_t$ ofnetwork nodes increases by increasing the coverage area. As the sensor spacing$d_n$ increases, the per-node information converges to its limit $D$ with rate$D-\sqrt{d_n}e^{-\alpha d_n}$ for a given diffusion rate $\alpha$. When thecoverage area is fixed and the node density increases, the per-node informationis inversely proportional to the node density. As the total energy $E_t$consumed in the network increases, the total information obtainable from thenetwork is given by $O(\log E_t)$ for the fixed node density and fixed coveragecase and by $\Theta (E_t^{2/3})$ for the fixed per-node sensing energy andfixed density and increasing coverage case.

How Much Information can One Get from a Wireless Ad Hoc Sensor Network over a Correlated Random Field?