Zipf's Law from Scale-free Geometry

TL;DR

This paper explains Zipf's law and population density fluctuations as consequences of scale-free clustering of humans on a two-dimensional surface, supported by empirical data and theoretical modeling.

Contribution

It introduces a theoretical framework linking scale-free spatial clustering and Zipf's law using random field theory, validated by empirical measurements.

Findings

Power spectrum slope measured at 2.04 ± 0.09

Theoretical prediction of slope as 2

Model enables new analytical predictions

Abstract

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.