We study the competition between random multiplicative growth and redistribution/migration in the mean-field limit, when the number of sites is very large but finite. We find that for static random growth rates, migration should be strong enough to prevent localization, i.e., extreme concentration on the fastest growing site. In the presence of an additional temporal noise in the growth rates, a third partially localized phase is predicted theoretically, using results from Derrida's random energy model. Such temporal fluctuations mitigate concentration effects, but do not make them disappear. We discuss our results in the context of population growth and wealth inequalities.

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• Competition
• Growth processes
• Phase transitions
• Disordered systems
• Population dynamics
• Social networks
• Langevin algorithm
• Mean field theory
• Statistical methods