Branching annihilating Levy flights: Irreversible phase transitions with long-range exchanges

The branching annihilation walk (BAW) process is generalized allowing either the particles or the offsprings (or both) to undergo Levy flights, where the flying length (l) is a random variable with a probability distribution given by P(1) proportional to l(-d-sigma) where sigma is the spatial dimension and a is the dimension of the random walk. The proposed branching annihilating Levy flights (BALF) process exhibits irreversible phase transitions between an active stationary state and a vacuum state. For short-ranged flights sigma > 1 the model belongs to the universality class of directed percolation (DP) as the standard BAW process. However, for smaller values of sigma, when the Levy walkers undergo long-range flights exhibiting superdiffusive behaviour, the critical exponents of the BALF model depart from those of DP indicating that the Levy mechanism effectively builds up additional long-range correlations. It is suggested. that transitions of the BALF model may belong to a superuniversality class of DP where the exponents are tunable functions of sigma.