Power-law and log-normal avalanche size statistics in random growth processes

TL;DR

This paper investigates avalanche size distributions in a minimal random growth model, revealing conditions under which they follow finite-scale, power-law, or log-normal statistics, supported by numerical and analytical analysis.

Contribution

It introduces a unified framework linking growth rate distributions to avalanche statistics, identifying the boundaries between different regimes.

Findings

Avalanche sizes follow finite-scale, power-law, or log-normal distributions depending on growth parameters.

Numerical simulations confirm the analytical predictions across various growth rate distributions.

The study provides a comprehensive phase diagram of avalanche regimes based on growth rate statistics.

Abstract

We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a and variance va. These two control parameters determine if the avalanche size tends to a stationary distribution, (Finite Scale statistics with finite mean and variance or Power-Law tailed statistics with exponent in (1, 3]), or instead to a non-stationary regime with Log-Normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by analytical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions and provide a precise definition of the boundaries between the three regimes.