  Associating the formation sites of haloes with the maxima of the smoothed
linear density field, we present non-perturbative predictions for the
Lagrangian and evolved halo correlation functions that are valid at all
separations. In Lagrangian space, we find significant deviations from the
perturbative bias calculation at small scales, in particular, a pronounced
exclusion region where $\xi=-1$ for maxima of unequal height. Our predictions
are in good agreement with the Lagrangian clustering of dark matter
proto-haloes reconstructed from N-body simulations. Our predictions for the
mean infall and velocity dispersion of haloes, which differ from the local bias
expansion, show a similar level of agreement with simulations. Finally, we
displace the initial density peaks according to the Zeldovich approximation in
order to predict the late-time clustering of dark matter haloes. While we are
able to reproduce the early evolution of this conserved set of tracers, our
approximation fails at the collapse epoch (z=0) on non-linear scales r<10Mpc/h,
emphasizing the need for a non-perturbative treatment of the halo displacement
field.
