  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.
