Most ecological theories that aim to explain coexistence in megadiverse communities employ a set of three rules to describe the stochastic geometry of biodiversity: (i) individuals exhibit intraspecific clustering; (ii) species abundances vary according to a log-normal distribution and (iii) the spatial arrangement between species is independent. The first two rules have received strong empirical support, but the third remains largely unexplored. To address this deficiency, we evaluated the independent species arrangement rule in a species-rich shrubland and its potential drivers, that is, the levels of species richness and intraspecific clustering exhibited by a given species at different scales, and the relative abundance of such species in the community. We found that interspecific associations were rare and that independence was positively related to species richness and intraspecific clustering, but negatively related to relative species abundances. Synthesis. Our results agree with the independent species arrangement rule and they provide empirical support for the stochastic geometry of biodiversity. In the context of species-rich plant communities, the likelihood of two species encountering is very small. However, our study demonstrated a novel feature of this context, where both intraspecific clustering (due limitations on dispersal) and relative species abundances play fundamental roles in determining the probability of two species encountering and interacting, especially at very fine spatial scales.