Determine the true growth rate in the partially localized phase

Determine the true asymptotic growth rate as a function of the redistribution rate in the partially localized phase of the mean-field growth model with quenched heterogeneous growth rates and multiplicative temporal noise, beyond the self-averaging approximation.

Background

The thesis identifies a partially localized phase in which a subextensive set of favorable sites contributes substantially to the total population, while the identity of the dominant site can change over time. The self-averaging treatment yields an estimate for the growth rate but is explicitly recognized as incomplete because fluctuations of the localized site are not self-averaging.

Finite-size numerical results suggest that an intermediate redistribution rate may maximize growth, whereas extrapolated large-system estimates appear to decrease with redistribution. The authors therefore leave the exact growth-rate function unresolved.

References

A remaining open question is to determine the true expression of $\gamma$ in the partially localized phase.

— Fluctuations and multifractality in stochastic models of interface growth and population dynamics  (2609.34468 - Bernard, 28 Sep 2026) in Section “Discussion,” Chapter “Mean-Field Growth with Quenched Heterogeneity and Multiplicative Noise”