  The correlation function xi(r) of matter in the non-linear regime is assumed
to be determined by the density profiles rho(r) and the mass distribution n(M)
of virialized halos. The Press--Schechter approach is used to compute n(M), and
the stable clustering hypothesis is used to determine the density profiles of
these Press--Schechter halos. Thus, the shape and amplitude of xi(r) on small
scales is related to the initial power spectrum of density fluctuations.
  The case of clustering from scale-free initial conditions is treated in
detail. If n is the slope of the initial power spectrum of density
fluctuations, then stable clustering requires that xi(r)\propto r^{-gamma},
where gamma is a known function of n. If halo--halo correlations can be
neglected, then rho(r)\propto r^{-epsilon}, where epsilon = (gamma+3)/2 =
3(4+n)/(5+n). For all values of n of current interest, this slope is steeper
than the value 3(3+n)/(4+n) that was obtained by Hoffman & Shaham in their
treatment of the shapes of the outer regions of collapsed halos. Our main
result is a prediction for the amplitude of the non-linear correlation
function. The predicted amplitude and its dependence on n are in good
quantitative agreement with N-body simulations of self-similar clustering.
  If stable clustering is a good approximation only inside the half-mass radii
of Press--Schechter halos, then the density contrast required for the onset of
stable clustering can be estimated. This density contrast is in the range
~300-600 and increases with the initial slope n, in agreement with estimates
from N-body simulations.
