Some exact results in branching and annihilating random walks

TL;DR

This paper derives exact results for branching and annihilating random walks, analyzing phase transitions and fixed point structures in different universality classes using a novel expansion method.

Contribution

It introduces an expansion around the Pure Annihilation model to obtain exact correlation functions and identifies the fixed point structure in the Parity Conserving class.

Findings

Exact correlation and response functions for the Pure Annihilation model.

Non-universal threshold for phase transition matches previous RG results.

Unstable PA fixed point in the Parity Conserving class across relevant dimensions.

Abstract

We present some exact results on the behavior of Branching and Annihilating Random Walks, both in the Directed Percolation and Parity Conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the non trivial Pure Annihilation model, whose correlation and response function we compute exactly. With this, the non-universal threshold value for having a phase transition in the simplest system belonging to the Directed Percolation universality class is found to coincide with previous Non Perturbative Renormalization Group approximate results. We also show that the Parity Conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.