On recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices

TL;DR

This paper generalizes Polya's recurrence theorem for fractional random walks on regular networks, analyzing their recurrence, transience, and passage times, revealing long-range step effects and non-local behavior.

Contribution

It introduces a fractional Laplacian-based random walk model, extending classical recurrence results to long-range step processes with Lévy flight characteristics.

Findings

Fractional random walks are recurrent or transient depending on the relation between dimension and alpha.

Explicit formulas for Green's functions and passage probabilities in 1D lattices.

Generalization of Polya's theorem to Lévy flight-based long-range steps.

Abstract

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type $L^{\frac{\alpha}{2}}$ where $L$ indicates a `simple' Laplacian matrix. We refer such walks to as `Fractional Random Walks' with admissible interval $0<\alpha \leq 2$. 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 Fractional Random Walk is transient for dimensions $d > \alpha$ (recurrent for $d\leq\alpha$) of the lattice. As a consequence for $0<\alpha< 1$ the Fractional Random Walk is transient for all lattice dimensions $d=1,2,..$ and in the range $1\leq\alpha < 2$ for dimensions $d\geq 2$. Finally, for $\alpha=2$ Polya's classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions $d\geq 3$. The generalization of Polya's recurrence theorem remains valid for the class of random walks with L\'evy flight asymptotics for long-range steps. We also analyze for the Fractional Random Walk mean first passage probabilities, mean first passage times, and global mean first passage times (Kemeny constant). For the infinite 1D lattice (infinite ring) we obtain for the transient regime $0<\alpha<1$ closed form expressions for the fractional lattice Green's function matrix containing the escape and ever passage probabilities. The ever passage probabilities fulfill Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the Fractional Random Walk is generated by the non-diagonality of the fractional Laplacian matrix with L\'evy type heavy tailed inverse power law decay for the probability of long-range moves.