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arXiv · 2610.22038

Partial Localization and Multifractality in Heterogeneous Population Growth with Resetting

Maximilien Bernard·Giuseppe Del Vecchio Del Vecchio·Jean-Philippe Bouchaud·Pierre Le Doussal
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Abstract

We study a model of heterogeneous multiplicative growth with mean-field redistribution and independent local stochastic resetting. The model describes populations or wealth distributed over sites with quenched growth-rate heterogeneity, while resets represent random local extinction or bankruptcy events. At large redistribution or resetting rate, we find a delocalized phase and we compute the growth rate as well as the stationary distribution of the population of a given site. We show that reset-free growth episodes generate power-law stationary tails with site-dependent exponents. For weak resetting and redistribution, favorable sites can undergo long reset-free episodes and localize. This produces a partially localized phase, in which a finite fraction of the total population is carried by a small set of sites whose identity changes on an $O(1)$ time scale within a subextensive pool of $N^{u^\star}$ favorable candidates, with $0<u^\star<1$. We derive the corresponding growth rate from a saddle point analysis, and show that the averaged stationary fractions display multifractal behavior.

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Maximilien Bernard, Giuseppe Del Vecchio Del Vecchio, Jean-Philippe Bouchaud, Pierre Le Doussal. 2026-10-15. Partial Localization and Multifractality in Heterogeneous Population Growth with Resetting. https://arxiv.org/abs/2610.22038

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