A simple unified view of branching process statistics: random walks in balanced logarithmic potentials

TL;DR

This paper presents a unified framework for understanding avalanche critical exponents across multiple universality classes by mapping their Langevin equations to a random walk in a balanced logarithmic potential, clarifying super-universality and external driving effects.

Contribution

It introduces a simple, unified perspective that links various universality classes to a common random walk model, emphasizing the role of external driving in critical behavior.

Findings

Unified view clarifies the relationship between different universality classes.

External driving influences the emergence of non-universal exponents.

Mapping to a random walk explains super-universality in avalanche statistics.

Abstract

We revisit the problem of deriving the mean-field values of avalanche critical exponents in systems with absorbing states. These are well-known to coincide with those of an un-biased branching process. Here, we show that for at least 4 different universality classes (directed percolation, dynamical percolation, the voter model or compact directed percolation class, and the Manna class of stochastic sandpiles) this common result can be obtained by mapping the corresponding Langevin equations describing each of these classes into a random walker confined close to the origin by a logarithmic potential. Many of the results derived here appear in the literature as independently derived for individual universality classes or for the branching process. However, the emergence of non-universal continuously-varying exponent values --which, as shown here, stems fro the presence of small external driving, that might induce avalanche merging-- has not been noticed (or emphasized) in the past. We believe that a simple an unified perspective as the one presented here can (i) help to clarify the overall picture, (ii) underline the super-universality of the behavior as well as the dependence on external driving, and (iii) help avoiding the common existing confusion between un-biased branching processes (equivalent to a random walker in a balanced logarithmic potential) and standard (un-confined) random walkers.