A mean-field theory for heterogeneous random growth with redistribution

TL;DR

The paper analyzes a mean-field model of heterogeneous random growth with redistribution, revealing how migration suppresses localisation and how temporal fluctuations induce a third, partially localised phase. By linking the static case to free cumulants and the R-transform, it derives a localisation transition driven by the upper edge of the growth-rate distribution. When temporal noise is added, a Derrida REM–type freezing mechanism predicts a rich three-phase diagram with delocalised, strongly localised, and partially localised regimes, including heavy-tailed allocations and finite-size effects. The results have implications for population dynamics and wealth inequality, illustrating how redistribution and luck jointly shape concentration and growth.

Abstract

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 localisation, i.e. extreme concentration on the fastest growing site. In the presence of an additional temporal noise in the growth rates, a third partially localised 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.

Figures (5)

• Figure 1: Main: Phase diagram predicted for a Gaussian distribution of $m_i$ in presence of noise $\sigma>0$, revealing three distinct phases. Phase I (delocalised with $\gamma=0$) & II (strongly localised with $\gamma_{\text{loc}} > 0$) are the continuation of the corresponding phases for $\sigma=0$. A new partially localised Phase III emerges, with a growth rate $\gamma_{\rm p.l.} = {\Sigma_0^2}/{2 \sigma^2}-\varphi > 0$, and a power-law distribution of the weights $p_{i}=x_i(t) /\bar{x}(t)$. The boundaries of phase I are $\varphi_c^{I-II}:= \Sigma_0 - {\sigma^2}/{2}$ and $\varphi_c^{I-III} := {\Sigma_0^2}/{2 \sigma^2}$. Inset: Plot of the growth rate $\gamma(\varphi)$ evaluated numerically (dots) in the absence of noise $\sigma=0$, for various $N$, with $m_i$ i.i.d Gaussian defined in text (with $\Sigma_0=1$). Our prediction (continuous line) is $\gamma = m_1 - \varphi$ in the strongly localised phase, while in the delocalised phase, one has footnote1$\gamma \approx \Sigma_0^2/(2 \varphi \, {\log N}) \to 0$. For $N \gg 1$, the transition occurs at $\varphi_c = m_1 - \varphi$footnotem1random.
• Figure 2: Main: Plot of the probability distribution of $p_{\rm max}=\max_i x_i/\sum_j x_j$ for various $N$ for $\varphi=0.2$ and $\sigma=0.3$. It shows that the system is localised for these values of the parameters. Inset: corresponding effective growth rates measured as $\langle \log{\frac{\bar{x}(t)}{\bar{x}(t_0)}} \rangle/(t-t_0)$ for $t_0=40$ (continuous line), compared to the prediction $\gamma= m_1 - \varphi - \frac{\sigma^2}{2}$ (dashed lines). The value of $m_1=m_1(N)$ converges to $\Sigma_0=1$ with logarithmic corrections.footnotefigpmax
• Figure S1: Numerical evaluation of $\bar{X}_N$ compared to the predicted value $X_\infty$ for $\Psi = 1.1$ and $\Psi = 2$ for several values of $N$. The parameters are set as $\varphi = 1$, $y = 1$, and $\tilde{\epsilon} = 1$. The results demonstrate that the difference between $\bar{X}_N$ and $X_\infty$ vanishes in the large $N$ limit,with a decay given by the lower cutoff $\sim N^{1-1/\Psi}$ (indicated by the black dashed line).
• Figure S2: Plot of $\gamma(\varphi)$ obtained numerically (dots), compared to the analytical prediction (continuous lines), for $\sigma=0$ and rigid $m_i$ given by the quantiles of $\rho(m)=\psi (1-m)^{\psi-1} \theta(0<m<1)$, for $\psi=1/2$ (no localised phase) and $\psi=2$.
• Figure S3: Left: Plot of $p_1$ in the absence of noise $\sigma = 0$ obtained numerically (dots) for $N=10^4$ with $m_i$ as rigid variables set by the quantiles of a Gaussian distribution with $\Sigma_0 = 1$. It is compared with our prediction, i.e. $p_1 =1 - \varphi/\varphi_c$ in the localised phase $\varphi<\varphi_c=m_1=\max_i m_i$, while it goes as $p_1 \sim 1/N$ in the delocalised phase. For $N=10^4$ one has $m_1 \approx 0.87$, which slowly converges to $1$ as $N$ further increases. Right: Main: Plot of the probability distribution of $p_{\rm max}=\max_i p_i$ for various $N$ for $\varphi=0.15$, $\Sigma_0=1$ and $\sigma^2=1.44$. It corresponds to the partially localised phase. The distribution decays as a power-law with exponent predicted by the REM argument $\mu= 1 -\frac{\Sigma_0^2}{\sigma^4}$. Inset: Measured growth rate $\gamma$ for $\sigma^2=2$ as a function of $\varphi$ for various $N$ (dots), compared to the asymptotic prediction $\gamma_{\rm p.l.}=\frac{\Sigma_0^2}{2 \sigma^2}-\varphi$ (dashed lines).