Zipf's Law from Scale-free Geometry
  The spatial distribution of people exhibits clustering across a wide range of
scales, from household ($\sim 10^{-2}$ km) to continental ($\sim 10^4$ km)
scales. Empirical data indicates simple power-law scalings for the size
distribution of cities (known as Zipf's law) and the population density
fluctuations as a function of scale. Using techniques from random field theory
and statistical physics, we show that these power laws are fundamentally a
consequence of the scale-free spatial clustering of human populations and the
fact that humans inhabit a two-dimensional surface. In this sense, the
symmetries of scale invariance in two spatial dimensions are intimately
connected to urban sociology. We test our theory by empirically measuring the
power spectrum of population density fluctuations and show that the logarithmic
slope $\alpha = 2.04 \pm 0.09$, in excellent agreement with our theoretical
prediction $\alpha = 2$. The model enables the analytic computation of many new
predictions by importing the mathematical formalism of random fields.
