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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