Branching and annihilating Lévy flights

We consider a system of particles undergoing the branching and annihilating reactions A-->(m+1)A and A+A-->∅, with m even. The particles move via long-range Lévy flights, where the probability of moving a distance r decays as r^-d-σ. We analyze this system of branching and annihilating Lévy flights using field theoretic renormalization group techniques close to the upper critical dimension d_c=σ with σ<2. These results are then compared with Monte Carlo simulations in d=1. For σ 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 σ bigger than about 3/2 in d=1, the critical branching rate moves away from zero with increasing σ, 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 σ.