New statistical tools for analyzing the structure of animal groups.

The statistical characterization of the spatial structure of large animal groups has been very limited so far, mainly due to a lack of empirical data, especially in three dimensions (3D). Here we focus on the case of large flocks of starlings (Sturnus vulgaris) in the field. We reconstruct the 3D positions of individual birds within flocks of up to few thousands of elements. In this respect our data constitute a unique set. We perform a statistical analysis of flocks' structure by using two quantities that are new to the field of collective animal behaviour, namely the conditional density and the pair correlation function. These tools were originally developed in the context of condensed matter theory. We explain what is the meaning of these two quantities, how to measure them in a reliable way, and why they are useful in assessing the density fluctuations and the statistical correlations across the group. We show that the border-to-centre density gradient displayed by starling flocks gives rise to an anomalous behaviour of the conditional density. We also find that the pair correlation function has a structure incompatible with a crystalline arrangement of birds. In fact, our results suggest that flocks are somewhat intermediate between the liquid and the gas phase of physical systems.