Large fluctuations and transport properties of the Lévy–Lorentz gas

The Levy-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 uniform 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, unexpectedly, the asymptotically stable diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion. Although the Levy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings lead to the conclusion that superdiffusion is a metastable behavior, which develops into normal diffusion on long timescales.