  The Geometrical Adhesion Model (GAM) we described in previous papers provides
a fully solved model for the nonlinear evolution of fields that mimic the
cosmological evolution of pressureless fluids. In this context we explore the
expected late time properties of the cosmic propagators once halos have formed,
in a regime beyond the domain of application of perturbation theories. Whereas
propagators in Eulerian coordinates are closely related to the velocity field
we show here that propagators defined in Lagrangian coordinates are intimately
related to the halo mass function. Exact results can be obtained in the 1D
case. In higher dimensions, the computations are more intricate because of to
the dependence of the propagators on the detailed shape of the underlying
Lagrangian-space tessellations, that is, on the geometry of the regions that
eventually collapse to form halos. We illustrate these results for both the 1D
and the 2D dynamics. In particular we give here the expected asymptotic
behaviors obtained for power-law initial power spectra. These analytical
results are compared with the results obtained with dedicated numerical
simulations.
