On the correlation structure of some random point processes on the line

The correlation structure of some remarkable point processes on the onedimensional real line is investigated. More specifically, focus is on translation invariant determinantal, permanental and/or renewal point processes. In some cases, anomalous (non-Poissonian) fluctuations for the number of points in a large window can be observed. This may be read from the total correlation function of the point process. We try to understand when and why this occurs and what are the anomalous behaviors to be expected. From examples, it is shown that determinantal (fermion) point processes can be super-homogeneous (the number variance grows slower than the number mean) and even hyper-uniform (when variance growth saturates). Renewal point processes with bounded spacings variance are essentially Poissonian (the number variance grows like the number mean as in Poisson models). Under certain conditions, permanental (boson) point processes can be subhomogeneous or critical (in the sense that the number variance grows faster than the number mean). We give several detailed examples illustrating these properties of interest together with unexpected behaviors.