Fractional random walk lattice dynamics

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 exhibit 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.