Equivalence of position-position auto-correlations in the Slicer Map and the L\'evy-Lorentz gas

The Slicer Map 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., CHAOS 25, 073113 (2015)] it was found that the moments of the position distributions as the Slicer Map have the same asymptotic behaviour as the L\'evy-Lorentz gas, a random walk on the line in which the scatterers are randomly distributed according to a L\'evy-stable probability distribution. Here we derive analytic expressions for the position-position correlations of the Slicer Map and, on the ground of this result, we formulate some conjectures about the asymptotic behaviour of position-position correlations of the L\'evy-Lorentz gas, for which the information in the literature is minimal. The numerically estimated position-position correlations of the L\'evy-Lorentz show a remarkable agreement with the conjectured asymptotic scaling.