T M Michelitsch , B A Collet , A P Riascos, A F Nowakowski and F C G A Nicolleau
Title: Fractional Random Walks on regular networks and lattices
We establish a generalization of Polya's recurrence theorem [1,2] 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 theorem remains valid for the class of random walks generated by Laplacian matrices having the same asymptotic power-law decay
as fractional Laplacian matrices.
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 closed form
expressions for the escape and ever passage probabilities holding in the transient regime $0<\alpha<1$. The ever passage probabilities fulfill
Riesz potential power law decay asymptotic behavior for nodes far from the departure node [3,4].
Our results
confirm our recent findings [3,4] that a search strategy based on a fractional random walk is faster than
a search strategy based on the normal random walk (α = 2)."
[1] G. Polya, Mathematische Annalen 83, 149 (1921).
[2] E.W. Montroll, J. SIAM 4/4, 241 (1956).
[3] T.M. Michelitsch, B.A. Collet, A.P Riascos, A. Nowakowski, F. Nicolleau, J Phys A 50, 50 (2017).
[4] A.P. Riascos, J.L. L. Mateos, Phys Rev E 90, 032809 (2014).