Large fluctuations and transport properties of the Lévy-Lorentz gas

Abstract

The Lévy-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 continuoustime 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 Lévy-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.

Discrete- and continuous-time random walks in 1D Lévy random medium

“…A recent preprint of Zamparo [12] claims the convergence of all quenched moments of X(t)/ √ t, under the additional assumption that the underlying RW is simple and symmetric (implying that X is the bona fide Lévy-Lorentz gas). Recall the notation m q for the q-th absolute moment of the standard Gaussian, cf.…”

“…Since X(t) is centered and scales like √ t, the probability of such "ballistic events" is expected to be exceedingly small. In [12,Thm. 2.3] it is proved that this probability vanishes like a stretched exponential.…”

“…Apart from recalling Corollary 3.2, which establishes the annealed CLT for X under the assumptions of [8] (µ < ∞ and S has symmetric, unimodal, finite-variance increments, cf. §2.1), in this section we present the results of [12] on the moments and large deviations of the annealed version of X.…”

“…A fair amount of mathematical work on the processes X and Y has been done in recent years by different authors [8,9,10,11,12]. In particular, in a number of cases, limit theorems have been proved for suitable rescalings of either process.…”

Ladder Costs for Random Walks in Lévy media

“…All these are processes that have been receiving a surge of attention as they model phenomena of anomalous transport and anomalous diffusion (see e.g. [16,28,3,20,29] for some general or recent references).…”