The universal macroscopic statistics and phase transitions of rank distributions

We establish a “Central Limit Theorem” for rank distributions, which provides a detailed characterization and classification of their universal macroscopic statistics and phase transitions. The limit theorem is based on the statistical notion of Lorenz curves, and is termed the “Lorenzian Limit Law” (LLL). Applications of the LLL further establish: (i) a statistical explanation for the universal emergence of Pareto’s law in the context of rank distributions; (ii) a statistical classification of universal macroscopic network topologies; (iii) a statistical classification of universal macroscopic socioeconomic states; (iv) a statistical classification of Zipf’s law, and a characterization of the “self-organized criticality” it manifests.