Finite-size scaling of survival probability in branching processes

TL;DR

This paper derives the finite-size scaling law for the survival probability in Galton-Watson branching processes, revealing universal behavior and providing exact scaling functions and critical exponents.

Contribution

It introduces the exact finite-size scaling law and critical exponents for survival probability in branching processes with finite offspring variance.

Findings

Derived the exact scaling function for survival probability.

Identified universal critical exponents.

Proved the universal behavior of branching processes.

Abstract

Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We reveal the finite-size scaling law of the survival probability for a given branching process ruled by a probability distribution of the number of offspring per element whose standard deviation is finite, obtaining the exact scaling function as well as the critical exponents. Our findings prove the universal behavior of branching processes concerning the survival probability.