I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process with these features, but I don't know the field very well and have had no success in my search thus far.
The most important property I want is for the points to be repulsive and in the sense that there is a characteristic distance $r> 0$ between any two nearby points, and attractive in the sense that it there is very rare to find zero probability of finding a ball of radius say, $10r$, in which there are no points. Finally, the process should be stationary so that the distribution is unchanged by translation. Isotropy (invariance under rotations) would be nice, but I don't really care. It is crucial for my purposes that it be an infinite process, defined on all of $\mathbb R^n$, and in dimension $n\geq 2$. I believe that in one dimension it is easy enough to construct such an example.
The idea is to model, for example, the location of trees in a forest.
Is there some well-known point process I am informally describing (or is it easy enough to construct one?), or is there some good reason I am having trouble finding one?