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Dynamic roughening of cities driven by multiplicative noise
Martin Hendrick1, Gabriele Manoli2
1Laboratory of Urban and Environmental Systems, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. martin.hendrick@epfl.ch.
Nature Communications
|June 25, 2026
Summary
Urban growth dynamics are modeled using geometric Brownian motion (GBM) with spatial coupling. This approach explains how local interactions shape urban morphology and roughness, linking city growth to statistical physics.
Area of Science:
- Urban studies
- Statistical physics
- Geomorphology
Background:
- Urban landscapes are rapidly changing, altering the planet's surface.
- Understanding urbanization and its spatiotemporal dynamics in the built environment is incomplete.
- Describing vertical and horizontal changes in urban structures presents a fundamental challenge.
Purpose of the Study:
- To model global building-height dynamics.
- To investigate the role of spatial correlations in urban morphology.
- To connect urban growth to established statistical-physics frameworks.
Main Methods:
- Modeling building-height dynamics using zero-dimensional geometric Brownian motion (GBM).
- Extending GBM with spatial coupling to incorporate intra-city correlations.
- Analyzing urban roughness using the Kardar-Parisi-Zhang (KPZ) equation in the continuum limit.
Main Results:
- Multiplicative noise in GBM explains stochastic fluctuations around economic growth.
- Spatial coupling mitigates noise-driven fluctuations, shaping urban morphology.
- Estimated roughness exponents for most cities fall within the Kardar-Parisi-Zhang (KPZ) range.
Conclusions:
- Multiplicative noise, moderated by local coupling, governs the evolution of urban roughness.
- City growth can be understood within a statistical-physics framework.
- The spatial model provides insights into the spatiotemporal evolution of urban landscapes.
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