Random walks with power-law fluctuations in the number of steps

In this paper we derive limiting distributions for the resultant of an anisotropic random walk, where the individual steps comprising the walk are drawn from distributions of arbitrary but finite variance, and where the number of steps fluctuate according to a discrete power-law distribution. The consideration of discrete distributions with power-law tails is motivated by their recently discovered relevance to complex systems and networks. When this random walk is unbiased, a power-law decay for large values of the amplitude of the resultant occurs. For a small directional bias, the power-law tail persists only in the direction parallel to the bias, exhibiting an exponential decay in all other directions. We consider the relevance of the limiting distributions to non-diffusive transport.