Ruelle (1969, Chapter 3) gives two conditions for an infinite-volume point process to have thermodynamic behaviour. The first condition temperedness is a restriction on the range of interactions between points. This will not concern us here, because we are interested only in finite point processes, for which the condition is irrelevant. Ruelle's second condition stability does apply to finite point processes.
This condition is clearly sufficient for h to be normalizable, since the normalizing constant is
Proposition 3.2.2 of Ruelle shows that this condition is sufficient for normalizability of densities determined by lower semicontinuous pair potentials, that is
where g is a lower semicontinuous function on 
.
The Strauss process does have a density of this form that violates the
stability condition since for each k there exist 
such
that s(x) = k (k - 1) / 2.
An even stronger stability condition is useful in establishing properties of Markov chain Monte Carlo simulation schemes for point processes. We shall assume the following stability condition.
This condition implies Condition 1 since it implies
It also implies a hereditary condition
which makes it possible to define the conditional intensity (see the contribution of Baddeley to this volume)
Thus Condition 2 can be restated saying that the process has a bounded conditional intensity.
The point of Condition 2 is that it permits easy proofs of geometric ergodicity of simple Markov chain Monte Carlo schemes for simulating the process. It is not clear that there are any interesting processes that satisfy Condition 1 and fail to satisfy Condition 2, but if there are they may require more delicate Markov chain Monte Carlo schemes or they may lead to a generalization of Condition 2 that still yields geometric ergodicity. Here we only investigate processes satisfying Condition 2.
