Lecture - online event

Abstract: We consider a one-dimensional process in random media that generalizes a model known in the physical literature as Levy-Lorentz gas. The media is provided by a renewal point process, while the dynamics is obtained as the linear interpolation of a random walk on the point process. We aim at investigating the annealed behavior of this process in the case in which the inter-distances between points and the length of the jumps of the random walk are i.i.d. stable random variables with parameters alpha and beta respectively. If beta =2 (finite variance of the random walk) we establish a functional limit theorem for the process, with a scaling limit that depends on the stability parameter alpha of the environment. In particular, we show that if alpha in (1,2) the system displays a diffusive behavior, while if alpha in (0,1) the behavior of the motion is super-diffusive with a scaling limit related to the so-called Kesten-Spitzer process. When beta is less than 2 (infinite variance of the random walk) we focus on the scaling behavior of the random walk on point process, showing different regimes of convergence depending on the range of alpha and beta. We finally discuss some works in progress about the 2-dimensional model, and present some related open questions .