Random Multiplicative Growth, Redistribution and Inequalities

Random multiplicative growth processes provide a simple yet remarkably powerful framework to understand a wide range of “scale‑free’’ phenomena, from city and firm sizes to wealth distributions. I will review how multiplicative noise generically generates Pareto (power‑law) tails. In the absence of redistribution (/migrations), the model reveals a genuine condensation transition: wealth/populations concentrates on a vanishing fraction of entities.

I will then introduce a generic stochastic model in which multiplicative growth is coupled to redistribution or diffusion on a network—motivated by migration between cities, wealth taxes and transfers, portfolio rebalancing, or species flow between habitats. This framework allows one to discuss (i) the asymptotic global growth rate, (ii) the tail exponent of the stationary distribution of “abundances”, and (iii) the conditions under which redistribution prevents condensation.

I will discuss the role of network topology, heterogeneity of local growth rates and their time-persistence, and show how these ingredients lead to non‑trivial “exploration–exploitation’’ trade‑offs and to an optimal tax or transfer rate. Connections with directed polymers, KPZ‑type growth, the Random Energy Model will be discussed, as well as recent applications in economics and ecology.