For the conditional Strauss process with k points, every direction is
a direction of recession (there just being two directions in the
one-dimensional parameter space). Taking the limit as 
gives the hard core process with k points. This is a
binomial process, k points uniformly distributed in the region,
conditioned on each pair of points having separation greater than r.
Taking the limit as 
gives the process conditioned
on the maximum value of s(x), which is obtained when every pair of points
are neighbours so s(x) = k (k - 1) / 2. This rather uninteresting process
has not been studied and has no name. Dub it the one-clump process.
In every realization of the process all k points are in a ball of diameter
r.
Even when the limit process is uninteresting in itself, it tells us something
about the original process. The reason why the conditional Strauss process
is an uninteresting model for clustering is that when 
all
realizations of the process look much like one of the limits 
or 
. The first is the binomial process and the
second is the one-clump process. At intermediate values of 
, the
conditional Strauss process is bimodal. Realizations in one mode look
much like realizations from the binomial process, slightly more clustered,
but hardly noticeable to the eye. Realizations in the other mode look like
realizations from the one-clump process, slightly less clustered, most but
not all of the points in a ball of diameter r. The parameter
only governs how much probability goes to each of the modes. This behaviour
of the Strauss process seems to have been first noticed by Julian Besag
(David Strauss, personal communication). A clear demonstration of this
bimodality can be seen in Figure 1 of Geyer and Thompson (1995).
It is interesting that the behaviour of the conditional Strauss process is so well described as a mixture of the processes obtained by taking limits in directions of recession. There is, of course, no reason why this phenomenon must always hold. There could be interesting novel behaviour at intermediate parameter values. Nevertheless, it does give a reason to think about all of the limiting processes. If none of the limiting processes are interesting, then the entire model may be uninteresting, as is the case for the Strauss process.