Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects

TL;DR

This paper explores the phase transition and finite-size effects in Brownian particles with drift and their mapping to branching processes, providing exact formulas and scaling laws for related probability distributions.

Contribution

It establishes an exact mapping between Brownian first-passage times and branching process sizes, revealing phase transition behavior and finite-size scaling laws.

Findings

Exact expressions for probability distribution Laplace transforms

Finite-size scaling laws for moments and distributions

Identification of a second-order phase transition in the system

Abstract

We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant boundary correspond to percolating trees, and those that do not correspond to non-percolating trees. In fact, both systems display a second-order phase transition between "insulating" and "conducting" phases, controlled by the drift velocity in the Brownian system. In the limit of large system size, we obtain exact expressions for the Laplace transforms of the probability distributions and their first and second moments. These quantities are also shown to obey finite-size scaling laws.