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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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Is there a natural random process that is rigorously known to produce Zipf's law?
Xavier Gabaix has several papers (e.g. here and here) that look at the emergence of Zipf's law in city sizes as a natural phenomenon.