L\'evy flights with power-law absorption

TL;DR

This paper analyzes how a particle performing Lévy flights on a one-dimensional lattice interacts with power-law distributed traps, revealing conditions under which the particle may never be trapped, supported by analytical and numerical results.

Contribution

It provides a novel analytical criterion for trapping probability in Lévy flights with power-law traps, extending understanding of search processes in complex environments.

Findings

Finite trapping probability when μ < α

Slow trapping times in finite systems

Extension to higher dimensions discussed

Abstract

We consider a particle performing a stochastic motion on a one-dimensional lattice with jump widths distributed according to a power-law with exponent $\mu + 1$. Assuming that the walker moves in the presence of a distribution $a(x)$ of targets (traps) depending on the spatial coordinate $x$, we study the probability that the walker will eventually find any target (will eventually be trapped). We focus on the case of power-law distributions $a(x) \sim x^{-\alpha}$ and we find that as long as $\mu < \alpha$ there is a finite probability that the walker will never be trapped, no matter how long the process is. This analytical result, valid on infinite chains, is corroborated by numerical simulations which also evidence the emergence of slow searching (trapping) times in finite-size system. The extension of this finding to higher-dimensional structures is also discussed.