Propagation and Extinction in Branching Annihilating Random Walks

TL;DR

This paper studies how branching annihilating random walks in one dimension evolve over time, focusing on the transition between sustained propagation and extinction, revealing unique critical behaviors and parity effects.

Contribution

It provides an exact solution for a specific case and develops an approximate analysis for general BAW processes, highlighting novel critical exponents and parity effects.

Findings

Exact solution for unit reaction probability case

Identification of parity-specific effects in propagation-extinction transition

Critical exponents differ from conventional models like Reggeon Field Theory

Abstract

We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite density wave, or extinction may occur, in which the number of particles vanishes in the long-time limit. The number parity conserving case where 2-offspring are produced in each branching event can be solved exactly for unit reaction probability, from which qualitative features of the transition between propagation and extinction, as well as intriguing parity-specific effects are elucidated. An approximate analysis is developed to treat this transition for general BAW processes. A scaling description suggests that the critical exponents which describe the vanishing of the particle density at the transition are unrelated to those of conventional models, such as Reggeon Field Theory. P. A. C. S. Numbers: 02.50.+s, 05.40.+j, 82.20.-w