Exact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process

TL;DR

This paper rigorously derives a finite-size scaling law and its corrections for the geometric Galton-Watson process, linking it to random walk behavior and clarifying the limits of scaling near criticality.

Contribution

It provides the first exact derivation of finite-size scaling laws and corrections for geometric Galton-Watson processes, including a mapping to random walks.

Findings

Finite-size scaling law is exactly derived for geometric Galton-Watson processes.

Corrections to the scaling law are explicitly calculated.

Results extend to random walks via a mapping, with the hitting probability as an order parameter.

Abstract

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.