  We compute precise predictions for the two-point correlation function of
local maxima (or minima) in the temperature of the microwave background, under
the assumption that it is a random gaussian field. For a given power spectrum
and peak threshold there are no adjustable parameters, and since this analysis
does not make the small-angle approximation of Heavens & Sheth (1999), it is
essentially complete. We find oscillatory features which are absent in the
temperature autocorrelation function, and we also find that the small-angle
approximation to the peak-peak correlation function is accurate to better than
0.01 on all scales. These high-precision predictions can form the basis of a
sensitive test of the gaussian hypothesis with upcoming all-sky microwave
background experiments MAP and Planck, affording a thorough test of the
inflationary theory of the early Universe. To illustrate the effectiveness of
the technique, we apply it to simulated maps of the microwave sky arising from
the cosmic string model of structure formation, and compare with the bispectrum
as a non-gaussian discriminant. We also show how peak statistics can be a
valuable tool in assessing and statistically removing contamination of the map
by foreground point sources.
