Fractional random walk lattice dynamics

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 fractional powers of Laplacian matrices L α 2 where α = 2 recovers the normal walk. First we demonstrate that the interval 0 < α ≤ 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 (α) , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix Z (α) 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 Lévy flights. As a further footprint of Lévy 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 − n α -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.

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

“…The long-range moves appearing in the FRW make the dynamics of the FRW remarkably rich. The present study is aiming to demonstrate some of these dynamic effects and complement some previous studies on the subject [6,28,29,30,31,32,33].…”

“…In this paper we have analyzed FRWs on regular networks, especially d-dimensional simple cubic lattices in the framework of Markovian processes. The FRW generalizes the Polya walk by replacing the Laplacian matrix L by a fractional power L α 2 with 0 < α ≤ 2 allowing within 0 < α < 2 long range moves where in sufficiently large networks the probability of occurence of long range steps decays as an inverse power law heavy tailed (Lévy-) distribution W This property is a landmark of the emergence of Lévy flights in sufficiently 'large' lattices [28,29,30].…”

“…11 A discussion and detailed proof can be found in [30]. 12 Defined on 0 ≤ τ < ∞ and decaying as τ → ∞ at least as f (τ ) ≤ const τ −β with β > α 2 as τ → ∞ This limiting expression for the fractional Laplacian matrix is universal for finite ergodic regular networks in the sense as it only requires one vanishing and N − 1 positive eigenvalues µ m independent of their values.…”

“…We identify the vanishing eigenvalue µ 1 = µ 0 = 0 of the stationary eigenvector p|Ψ 0 = 1 √ N corresponding to the zero (Bloch-wave) vector κ = (0, .., 0). We see from this relation that such matrices are of the form B p| q = B( p − q) reflecting translational invariance, and all N × N -matrices fulfill N j -periodic boundary conditions B( s) = B(s 1 , .., s j , .., s d ) = B(s 1 , .., s j + N j , .., s d ) in all dimensions j = 1, .., d and further symmetries such as (generalized) Töplitz structure have been outlined elsewhere [30,31,34]. Further useful is the transition to infinite lattices when all N j → ∞ which we write compactly 13…”

Random walks with long-range steps generated by functions of Laplacian matrices

“…(46) is revealed. In addition, by using the methods developed in [27] for the power function, different asymptotic results can be deduced for n-dimensional lattices.…”