  The detection of hidden two-dimensional Gauss-Markov random fields using
sensor networks is considered. Under a conditional autoregressive model, the
error exponent for the Neyman-Pearson detector satisfying a fixed level
constraint is obtained using the large deviations principle. For a symmetric
first order autoregressive model, the error exponent is given explicitly in
terms of the SNR and an edge dependence factor (field correlation). The
behavior of the error exponent as a function of correlation strength is seen to
divide into two regions depending on the value of the SNR. At high SNR,
uncorrelated observations maximize the error exponent for a given SNR, whereas
there is non-zero optimal correlation at low SNR. Based on the error exponent,
the energy efficiency (defined as the ratio of the total information gathered
to the total energy required) of ad hoc sensor network for detection is
examined for two sensor deployment models: an infinite area model and and
infinite density model. For a fixed sensor density, the energy efficiency
diminishes to zero at rate O(area^{-1/2}) as the area is increased. On the
other hand, non-zero efficiency is possible for increasing density depending on
the behavior of the physical correlation as a function of the link length.
