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We investigate front propagation in systems with diffusive and sub-diffusivebehavior. The scaling behavior of moments of the diffusive problem, both in thestandard and in the anomalous cases, is not enough to determine the features ofthe reactive front. In fact, the shape of the bulk of the probabilitydistribution of the transport process, which determines the diffusiveproperties, is important just for pre-asymptotic behavior of front propagation,while the precise shape of the tails of the probability distribution determinesasymptotic behavior of front propagation.

Linear and anomalous front propagation in system with non Gaussian diffusion: the importance of tails