Front propagation in reaction-superdiffusion dynamics: Taming Levy flights with fluctuations

We investigate front propagation in a reacting particle system in which particles perform scale-free random walks known as Levy flights. The system is described by a fractional generalization of a reaction-diffusion equation. We focus on the effects of fluctuations caused by a finite number of particles per volume. We show that, in spite of superdiffusive particle dispersion and contrary to mean-field theoretical predictions, wave fronts propagate at constant velocities, even for very large particle numbers. We show that the asymptotic velocity scales with the particle number and obtain the scaling exponent.