Equivalence of position–position auto-correlations in the Slicer Map and the Lévy–Lorentz gas

The Slicer Map (SM) is a one-dimensional non-chaotic dynamical system that shows sub-, super-, and normal diffusion as a function of its control parameter. In a recent paper (Salari et al 2015 Chaos 25 073113) it was found that the moments of the position distributions as the SM have the same asymptotic behaviour as the Lévy–Lorentz gas (LLg), a random walk on the line in which the scatterers are randomly distributed according to a Lévy-stable probability distribution. Here we derive analytic expressions for the position–position correlations of the SM and, on the ground of this result, we formulate some conjectures about the asymptotic behaviour of position–position correlations of the LLg, for which the information in the literature is minimal. The numerically estimated position–position correlations of the Lévy–Lorentz show a remarkable agreement with the conjectured asymptotic scaling