As with the unconditional Strauss process, the direction of recession
gives the hard core process as a limiting distribution.
The upper bound 
gives a new limiting process. Consider
the direction 
. Then
and is maximized on the set
the set of points such that each point has at least c neighbours. The
limiting process is the family of Poisson processes conditioned to lie
in 
.
Directions of recession with 
and 
give rise to the uninteresting limit distribution conditioned on the empty
realization. Directions of recession with 
give
rise to no limit distributions since, is empty.
As with the unconditional Strauss process, directions of recession with
and seem difficult to describe.
They are, at least models, for repulsion rather than clustering. So we
can say that the saturation process has very simple limiting behaviour
in the clustering region of the parameter space.
No such simple limits arise with the triplets process. Not only is the parameter space three-dimensional so that we must consider directions of recession in three dimensions, there are no simple linear inequalities involving the canonical statistics n(x), s(x) and w(x). All of the limiting behaviour of the triplets process seems complicated. This is one way to see that the triplets process, despite its simple motivation from the point of view of cliques in Markov point processes, is actually a much more complicated statistical model than the saturation process.