3:15 pm: Alessandra Bianchi, Università degli Studi di Padova

Random walks on a Lévy-type random media

We consider a one-dimensional process in random media that generalizes the model known in the physical literature as Lévy-Lorentz gas. The medium is provided by a renewal point process in which the inter-distances between points are i.i.d. heavy-tailed random variables, while the dynamics is obtained as the linear interpolation of - possibly long jump - random walks on the point process. These models have been used to describe phenomena that exhibit superdiffusion, and the main focus of this investigation is on the derivation of the scaling behavior of the process as a function of the parameters that enter its definition. We give an account on a number of recent theorems, which include non-standard functional limit theorems for the process in discrete and continuous time, and the distributional characterization of the first-passage time. We conclude by discussing the two-dimensional setting and some related open problems.

4:15 pm Coffee break

4:45 pm: Peter Mörters, University of Cologne

Condensation in scale-free geometric graphs with excess edges

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with heavy-tailed radius distribution, and the age-dependent random connection model. In all these cases the mechanism behind the large deviation is based on a condensation effect for the vertex degrees. The mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We also study the empirical distribution of edge lengths under the conditioning, which splits into a bulk and travelling wave part of asymptotically positive proportions. The talk is based on joint work with Remco van der Hofstad, Pim van der Hoorn, Céline Kerriou and Neeladri Maitra.