  We use numerical simulations of a (480 Mpc/h)^3 volume to show that the
distribution of peak heights in maps of the temperature fluctuations from the
kinematic and thermal Sunyaev-Zeldovich effects will be highly non-Gaussian,
and very different from the peak height distribution of a Gaussian random
field. We then show that it is a good approximation to assume that each peak in
either SZ effect is associated with one and only one dark matter halo. This
allows us to use our knowledge of the properties of haloes to estimate the peak
height distributions. At fixed optical depth, the distribution of peak heights
due to the kinematic effect is Gaussian, with a width which is approximately
proportional to optical depth; the non-Gaussianity comes from summing over a
range of optical depths. The optical depth is an increasing function of halo
mass, and the distribution of halo speeds is Gaussian, with a dispersion which
is approximately independent of halo mass. This means that observations of the
kinematic effect can be used to put constraints on how the abundance of massive
clusters evolves, and on the evolution of cluster velocities. The
non-Gaussianity of the thermal effect, on the other hand, comes primarily from
the fact that, on average, the effect is larger in more massive haloes, and the
distribution of halo masses is highly non-Gaussian. We also show that because
haloes of the same mass may have a range of density and velocity dispersion
profiles, the relation between halo mass and the amplitude of the thermal
effect is not deterministic, but has some scatter.
