  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.
