arXiv subjects

B. A. Collet

Publications and source records attributed to B. A. Collet.

3 recordsLinked to original sources

arXiv2018

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

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions Lévy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions. In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like Lévy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks.

cond-mat.stat-mech↗

arXiv2017

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

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type $L^{\fracα{2}}$ where $L$ indicates a `simple' Laplacian matrix. We refer such walks to as `Fractional Random Walks' with admissible interval $0<α\leq 2$. 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 $d > α$ (recurrent for $d\leqα$) of the lattice. As a consequence for $0<α< 1$ the Fractional Random Walk is transient for all lattice dimensions $d=1,2,..$ and in the range $1\leqα< 2$ for dimensions $d\geq 2$. Finally, for $α=2$ 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 Lévy 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 $0<α<1$ 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 Lévy type heavy tailed inverse power law decay for the probability of long-range moves.

cond-mat.stat-mech↗

arXiv2016

A fractional generalization of the classical lattice dynamics approach

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and infinite lattice in n=1,2,3,..n=1,2,3,.. dimensions. The present model which is based on Hamilton's variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.

math-ph↗