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Mean-field theory for heterogeneous random growth with redistribution
Maximilien Bernard1,2, Jean-Philippe Bouchaud3, Pierre Le Doussal1
1Université Paris Cité, Sorbonne Université, ENS & PSL University, Laboratoire de Physique de l'École Normale Supérieure, CNRS, 75005 Paris, France.
Physical Review. E
|April 18, 2026
Summary
Migration dynamics are crucial for preventing extreme concentration in growing systems. Temporal noise in growth rates can lead to partial localization, impacting population growth and wealth inequality.
Area of Science:
- Statistical physics
- Complex systems modeling
Background:
- Understanding the interplay between growth and redistribution is key in various complex systems.
- Mean-field theory provides a framework for analyzing large, interacting systems.
Purpose of the Study:
- To investigate the competition between random multiplicative growth and redistribution/migration.
- To analyze the impact of static and temporally fluctuating growth rates on system localization.
- To explore implications for population dynamics and wealth distribution.
Main Methods:
- Mean-field limit analysis for a large but finite number of sites.
- Application of concepts from Derrida's random energy model for temporal noise.
- Theoretical prediction of system phases.
Main Results:
- Sufficiently strong migration prevents localization under static growth rates.
- Temporal noise introduces a partially localized phase, mitigating but not eliminating concentration.
- Concentration effects persist despite temporal fluctuations.
Conclusions:
- Migration is essential for mitigating extreme concentration in growing systems.
- Temporal noise in growth rates creates complex dynamics with partial localization.
- The findings offer insights into mechanisms driving population growth and wealth inequalities.
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