Lévy flights in a quenched jump length field: a 1-loop renormalization group approach

Transmission probability through a Lévy glass and comparison with a Lévy walk

Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power-law distribution of radii (a so-called Lévy glass) have found that the transmission probability T∝1/L(γ) scales superdiffusively (γ<1). The data has been interpreted in terms of a Lévy walk. We present computer simulations to demonstrate that diffusive scaling (γ≈1) can coexist with a divergent second moment of the step size distribution [p(s)∝1/s(1+α) with α<2]. This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a Lévy walk of independent steps.

Role of quenching on superdiffusive transport in two-dimensional random media

Transport in random media is known to be affected by quenched disorder. From the point of view of random walks, quenching induces correlations between steps that may alter the dynamical properties of the medium. This paper is intended to provide more insight into the role of quenched disorder on superdiffusive transport in two-dimensional random media. The systems under consideration are disordered materials called Lévy glasses that exhibit large spatial fluctuations in the density of scattering elements. We show that in an ideal Lévy glass the influence of quenching can be neglected, in the sense that transport follows to very good approximation that of a standard Lévy walk. We also show that, by changing sample parameters, quenching effects can be increased intentionally, thereby making it possible to investigate systematically diverse regimes of transport. In particular, we find that strong quenching induces local trapping effects which slow down superdiffusion and lead to a transient subdiffusivelike transport regime close to the truncation time of the system.