Fractional random walk lattice dynamics

TL;DR

This paper investigates fractional random walks on regular lattices, deriving analytical expressions for their dynamics, revealing Le9vy flight behavior, and showing how fractional walks enhance exploration efficiency compared to normal walks.

Contribution

It introduces a comprehensive analytical framework for fractional random walks on lattices, connecting them with Le9vy flights and small world properties, extending prior work on normal random walks.

Findings

Fractional transition matrices exhibit power law decay indicating Le9vy flights.

Long-time behavior dominated by slow relaxing modes with t^{-rac{n}{\alpha}} decay.

Fractional walks increase exploration efficiency due to long-range moves.

Abstract

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.