  The origin(s) of the ubiquity of Zipf's law (an inverse power law form for
the probability density function (PDF) with exponent $1+1$) is still a matter
of fascination and investigation in many scientific fields from linguistic,
social, economic, computer sciences to essentially all natural sciences. In
parallel, self-excited dynamics is a prevalent characteristic of many systems,
from seismicity, financial volatility and financial defaults, to sociology,
consumer behaviors, computer sciences, The Internet, neuronal discharges and
spike trains in biological neuron networks, gene expression and even
criminology. Motivated by financial and seismic modelling, we bring the two
threads together by introducing a general class of nonlinear self-excited point
processes with fast-accelerating intensities as a function of "tension".
Solving the corresponding master equations, we find that a wide class of such
nonlinear Hawkes processes have the PDF of their intensities described by
Zipf's law on the condition that (i) the intensity is a fast-accelerating
function of tension and (ii) the distribution of the point fertilities is
symmetric. This unearths a novel mechanism for Zipf's law, providing a new
understanding of its ubiquity.
