Large fluctuations and transport properties of the L\'evy-Lorentz gas

TL;DR

This paper investigates the large fluctuation behavior and transport properties of the Levy-Lorentz gas, revealing that superdiffusion is transient and normal diffusion dominates in typical configurations over long times.

Contribution

It provides a rigorous large deviation principle for annealed fluctuations and asymptotics of moments, clarifying the transition from superdiffusion to normal diffusion in the model.

Findings

Annealed fluctuations exhibit superdiffusion with precise large deviation asymptotics.

Quenched fluctuations show normal diffusion, indicating typical configurations lead to diffusive behavior.

Superdiffusion appears transient, transitioning to normal diffusion over long timescales.

Abstract

The L\'evy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a precise large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that the asymptotic diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion, and not superdiffusion. Although the L\'evy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings suggest that superdiffusion is a transient behavior which develops into normal diffusion on long timescales, and raise a new question about how the transition from the quenched normal diffusion to the annealed superdiffusion occurs.