The neutral theory of biodiversity attributes community structure to the effects of chance alone, assuming that all species and individuals are demographically equivalent. Here we present a spatially explicit version of the neutral theory and test it against the Barro Colorado Island (BCI) data. Monitoring the dynamics of clusters, we show that the effect of local heterogeneities (e.g., microtopography) is weak, making a spatially homogenous model plausible. We then compare the cluster statistics of the three most frequent species with the patterns obtained from neutral dynamics, examining two families of recruitment kernels: one that interpolates between a limited distance and panmictic dispersal (local-global) and one that assumes a scale-free Cauchy kernel. The results rule out the local-global dispersal model and show that the spatial patterns fit very nicely those obtained from the fat-tailed kernel. Our work emphasizes the importance of spatiotemporal cluster dynamics as an instrument for detecting the factors that govern community assembly.