The spatial distribution of people exhibits clustering across a wide range of scales , from household ( \sim 10 ^ { -2 } \text { km } ) to continental ( \sim 10 ^ { 4 } \text { 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 .