Michelitsch, T.M., Collet, B.A., Riascos, A.P. et al. (2 more authors) (2017) Fractional random walk lattice dynamics. Journal of Physics A: Mathematical and Theoretical, 50 (5). 055003055003. ISSN 17518113
Abstract
We analyze timediscrete and timecontinuous 'fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in n = 1, 2, 3,.. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices ${{L}^{\frac{\alpha}{2}}}$ where $\alpha =2$ recovers the normal walk. First we demonstrate that the interval $0<\alpha \leqslant 2$ is admissible for the fractional random walk. We derive analytical expressions for the transition matrix of the fractional random walk and closely related the average return probabilities. We further obtain the fundamental matrix ${{Z}^{(\alpha )}}$ , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix ${{Z}^{(\alpha )}}$ relates fractional random walks with normal random walks. We show that the matrix elements of the transition matrix of the fractional random walk exihibit for large cubic ndimensional lattices a power law decay of an ndimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the ndimensional space, the transition matrix and return probabilities of the fractional random walk are dominated for large times t by slowly relaxing longwave modes leading to a characteristic ${{t}^{\frac{n}{\alpha}}}$ decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.
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Copyright, Publisher and Additional Information:  © 2017 IOP Publishing. This is an author produced version of a paper subsequently published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's selfarchiving policy. 
Dates: 

Institution:  The University of Sheffield 
Academic Units:  The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Mechanical Engineering (Sheffield) 
Depositing User:  Symplectic Sheffield 
Date Deposited:  04 Apr 2017 14:48 
Last Modified:  06 Jan 2018 01:38 
Published Version:  http://doi.org/10.1088/17518121/aa5173 
Status:  Published 
Publisher:  IOP Publishing 
Refereed:  Yes 
Identification Number:  https://doi.org/10.1088/17518121/aa5173 