Zipf's law emerges asymptotically during phase transitions in communicative systems

TL;DR

This paper demonstrates that Zipf's law naturally emerges during phase transitions in language communication systems, with power-law scaling arising asymptotically at high word ranks due to the underlying mathematical properties of the transition.

Contribution

It provides a mathematical explanation for the emergence of Zipf's law during phase transitions in communicative systems, linking it to the Laplace transform of the step function.

Findings

Zipf's law appears asymptotically at high word ranks during phase transitions.

The emergence of Zipf's law is explained by the Laplace transform of the step function.

Rare words play a critical role in effective communication at the phase transition.

Abstract

Zipf's law predicts a power-law relationship between word rank and frequency in language communication systems, and is widely reported in texts yet remains enigmatic as to its origins. Computer simulations have shown that language communication systems emerge at an abrupt phase transition in the fidelity of mappings between symbols and objects. Since the phase transition approximates the Heaviside or step function, we show that Zipfian scaling emerges asymptotically at high rank based on the Laplace transform. We thereby demonstrate that Zipf's law gradually emerges from the moment of phase transition in communicative systems. We show that this power-law scaling behavior explains the emergence of natural languages at phase transitions. We find that the emergence of Zipf's law during language communication suggests that the use of rare words in a lexicon is critical for the construction of an effective communicative system at the phase transition.