Critical spreading dynamics of parity conserving annihilating random walks with power-law branching

We investigate the critical spreading of the parity conserving annihilating random walks model with Lévy-like branching. The random walks are considered to perform normal diffusion with probability p on the sites of a one-dimensional lattice, annihilating in pairs by contact. With probability 1−p, each particle can also produce two offspring which are placed at a distance r from the original site following a power-law Lévy-like distribution P(r)∝1∕rα. We perform numerical simulations starting from a single particle. A finite-time scaling analysis is employed to locate the critical diffusion probability pc below which a finite density of particles is developed in the long-time limit. Further, we estimate the spreading dynamical exponents related to the increase of the average number of particles at the critical point and its respective fluctuations. The critical exponents deviate from those of the counterpart model with short-range branching for small values of α. The numerical data suggest that continuously varying spreading exponents sets up while the branching process still results in a diffusive-like spreading.