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

New large deviations results that characterize the asymptotic information rates for general $d$-dimensional ($d$-D) stationary Gaussian fields are obtained. By applying the general results to sensor nodes on a two-dimensional (2-D) lattice, the asymptotic behavior of ad hoc sensor networks deployed over correlated random fields for statistical inference is investigated. Under a 2-D hidden Gauss-Markov random field model with symmetric first order conditional autoregression and the assumption of no in-network data fusion, the behavior of the total obtainable information [nats] and energy efficiency [nats/J] defined as the ratio of total gathered information to the required energy is obtained as the coverage area, node density and energy vary. When the sensor node density is fixed, the energy efficiency decreases to zero with rate $\Theta({area}^{-1/2})$ and the per-node information under fixed per-node energy also diminishes to zero with rate $O(N_t^{-1/3})$ as the number $N_t$ of network 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 the coverage area is fixed and the node density increases, the per-node information is inversely proportional to the node density. As the total energy $E_t$ consumed in the network increases, the total information obtainable from the network is given by $O(\log E_t)$ for the fixed node density and fixed coverage case and by $\Theta (E_t^{2/3})$ for the fixed per-node sensing energy and fixed density and increasing coverage case.