Field-theoretic approach to the universality of branching processes

TL;DR

This paper develops a field-theoretic framework to analyze continuous-time branching processes, revealing their universal scaling behavior and showing that only the first two moments of the offspring distribution are needed near criticality.

Contribution

The authors derive a field theory for branching processes and demonstrate its use in calculating universal observables and their scaling, highlighting the sufficiency of the first two moments.

Findings

Universal scaling of moments, survival probability, and avalanche shape.

First and second moments of offspring distribution suffice near criticality.

Analytical results confirmed by computer simulations.

Abstract

Branching processes are widely used to model phenomena from networks to neuronal avalanching. In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the $n$-point correlation function and the probability density function of the total avalanche size. Previous studies have shown universality in certain observables of branching processes using probabilistic arguments, however, a comprehensive description is lacking. We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size. Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterise the process in the vicinity of criticality, regardless of the underlying offspring distribution. This finding implies that branching processes are universal. We illustrate our analytical results with computer simulations.