2017
DOI: 10.1088/1751-8121/aa9008
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Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices

Abstract: We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type L α 2 where L indicates a 'simple' Laplacian matrix. We refer such walks to as 'Fractional Random Walks' with admissible interval 0 < α ≤ 2. We deduce for the Fractional Random Walk probability generating functions (network Green's functions). From these analytical results we establish a generalization of Polya's recurrence theorem for Fractional Random Walks on d-dimensional infinite lattices: The Fra… Show more

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Cited by 27 publications
(56 citation statements)
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“…We refer the resulting diffusion process to as 'generalized fractional diffusion'. The eigenvalues of the one-step transition matrix (55) for the d-dimensional infinite lattice are given by the Fourier transforms [33,34]…”
Section: Generalized Fractional Diffusion In Z Dmentioning
confidence: 99%
“…We refer the resulting diffusion process to as 'generalized fractional diffusion'. The eigenvalues of the one-step transition matrix (55) for the d-dimensional infinite lattice are given by the Fourier transforms [33,34]…”
Section: Generalized Fractional Diffusion In Z Dmentioning
confidence: 99%
“…, N . The topology of the network is described by the positive-semidefinite N × N Laplacian matrix which is defined by [8,12,36,37,38,39,40] L…”
Section: Continuous-time Random Walk On Networkmentioning
confidence: 99%
“…For the random walk on the network we allow long-range jumps which can be described when we replace the Laplacian matrix by its fractional power in the one-step transition matrix (45). In this way, the walker cannot only jump to connected neighbor nodes, but also to far distant nodes in the network [5,6,8,34,36,37,38,39,42,43]. The model to be developed in this section involves both space-and time-fractional calculus.…”
Section: Generalized Space-time Fractional Diffusion In Z Dmentioning
confidence: 99%
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