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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2017

Fractional random walk lattice dynamics

Dynamique fractionnaire des marches aleatoires

Résumé

We analyze time-discrete and continuous `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 {\it fractional powers of Laplacian matrices $L^{\frac{\alpha}{2}}$} where $\alpha=2$ recovers the normal walk. First we demonstrate that the interval $0<\alpha\leq 2$ is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix 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 fractional transition matrix elements exihibit for large cubic $n$-dimensional lattices a power law decay of an $n$-dimensional infinite space Riesz fractional derivative type indicating emergence of L\'evy flights. As a further footprint of L\'evy flights in the $n$-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times $t$ by slowly relaxing long-wave 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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Dates et versions

hal-01357909 , version 1 (30-08-2016)

Identifiants

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Thomas M. Michelitsch, Bernard Collet, Alejandro Perez Riascos, Andrzeij Nowakowski, Franck C.G.A Nicolleau. Fractional random walk lattice dynamics. Journal of Physics A: Mathematical and Theoretical, 2017, J. Phys. A: Math. Theor. 50, pp.055003 ⟨10.1088/1751-8121/aa5173⟩. ⟨hal-01357909⟩
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