Branching and annihilating Levy flights

Abstract

We consider a system of particles undergoing the branching and annihilating reactions A -> (m+1)A and A + A -> 0, with m even. The particles move via long-range Levy flights, where the probability of moving a distance r decays as r^{-d-sigma}. We analyze this system of branching and annihilating Levy flights (BALF) using field theoretic renormalization group techniques close to the upper critical dimension d_c=sigma, with sigma<2. These results are then compared with Monte-Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an active phase occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the critical branching rate moves smoothly away from zero with increasing sigma, and the transition lies in a different universality class, inaccessible to controlled perturbative expansions. We measure the exponents in both universality classes and examine their behavior as a function of sigma.

9 pages, 4 figures

Cited by in corpus (18)

• Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems
• Non-equilibrium Phase Transitions with Long-Range Interactions
• Criticality of Spin Systems with Weak Long-Range Interactions
• Long Range Hops and the Pair Annihilation Reaction A+A->0: Renormalization Group and Simulation
• Competition between local erasure and long-range spreading of a single biochemical mark leads to epigenetic bistability
• Some exact results in branching and annihilating random walks
• Reaction, Levy Flights, and Quenched Disorder
• Branching rate expansion around annihilating random walks
• Universality classes of absorbing phase transitions in generic branching-annihilating particle systems
• Directed percolation with incubation times
• Fractional Generalization of Kac Integral
• Survival in two-species reaction-superdiffusion system: Renormalization group treatment and numerical simulations
• The pair annihilation reaction D + D --> 0 in disordered media and conformal invariance
• Robustness of first-order phase transitions in one-dimensional long-range contact processes
• Crossover behaviors in branching annihilating attracting walk
• Branching annihilating random walks with long-range attraction in one dimension
• Critical dynamics in systems controlled by fractional kinetic equations
• Fluctuation Effects in the Pair-Annihilation Process with Lévy Dynamics