Efficient approach to time-dependent super-diffusive Lévy random walks on finite 2D-tori using circulant analogues

Abstract

This work resumes the investigation on discrete-time super-diffusive in Lévy random walks defined on networks by using a inverse problem approach, with a focus on 2D-tori. Imposing that the mean square displacement of the walker should be proportional to tγ, we use a Markov Chain formalism to evaluate a fine tuned time-dependent probability distribution of long-distance jumps the walker should use to meet this dependency. Despite its wide applicability, calculations are time-intensive, with a computing time proportional to the number of nodes in the graph to a power >3.4. Here it is shown that, by using the circulant property satisfied by the adjacency matrices of a class of tori, it is possible to significantly speed up the calculations. For the purpose of comparison, the inverse super-diffusion problem is solved for two tori based on finite patches of the two-dimensional square lattice, namely the usual (non-circulant) and the helical (circulant) ones. The results of the latter, based on derived new expressions to compute the mean square displacement valid for circulant tori, are in complete agreement with those derived using general expressions, even if the computing time increases with respect to the number of nodes with a significantly smaller exponent ≳2.1. Numerical simulations in both tori types also reproduce super-diffusion when using the time-dependent probability distributions obtained for the helical one. The results suggest that this time efficient approach can be extended to model super-diffusion on cubic and hyper-cubic lattices.