Theory of Branching and Annihilating Random Walks

TL;DR

This paper develops a comprehensive field-theoretic framework for diffusion-limited reactions involving branching and annihilating particles, revealing phase transitions and universality classes depending on particle interaction rules and spatial dimensions.

Contribution

It introduces a systematic renormalization group approach to analyze branching-annihilating random walks, identifying critical dimensions and phase transition behaviors for different particle interaction cases.

Findings

For even m, a nontrivial transition appears below a critical dimension.

For odd m, a phase transition exists for all dimensions up to 2.

The transition for odd m belongs to the directed percolation universality class.

Abstract

A systematic theory for the diffusion--limited reaction processes $A + A \to 0$ and $A \to (m+1) A$ is developed. Fluctuations are taken into account via the field--theoretic dynamical renormalization group. For $m$ even the mean field rate equation, which predicts only an active phase, remains qualitatively correct near $d_c = 2$ dimensions; but below $d_c' \approx 4/3$ a nontrivial transition to an inactive phase governed by power law behavior appears. For $m$ odd there is a dynamic phase transition for any $d \leq 2$ which is described by the directed percolation universality class.