We introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge- and to vertex-reinforced random walks with super-linear reinforcement on arbitrary in infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs.

For edge-reinforced random walks, we complete the results of Limic and Tarrès [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 94-26, Purdue University (1994)] by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times.

For vertex-reinforced random walks, we extend results previously obtained on $\mathbb{Z}$ by Volkov [Ann. Probab. (2001)] and by Basdevant, Schapira and Singh [Ann. Probab. (2014)], and on complete graphs by Benaim, Raimond and Schapira [ALEA (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.

This is joint work with Debleena Thacker.

We will discuss a recent model of random geometric graphs on the hyperbolic plane that was introduced by Krioukov et al. as a model for complex networks. This talk will focus on the typical component structure of this model in terms of its basic parameters. We will also consider the robustness of these random graphs as well as the evolution of bootstrap percolation processes on them. We will show that in a certain range of the parameters of the model, metastability-type phenomena emerge.

In this joint work with Steffen Dereich and Peter Mörters, we introduce and study a branching process with both a reinforcement and a mutation dynamics. Our model incompasses as a particular case the Bianconi and Barabási's preferential attachment graph with fitnesses but also a selection and mutation population branching process. We prove that this model exhibits, under some conditions on the parameters, a condensation phenomenon, and more precisely that ``the winner does not take it all'', disproving a claim made in the physics literature about the Bianconi and Barabási model.

Plants constantly adapt their growth orientations by bending in response to environmental cues, including light, gravity and more. We study the response of roots to gravity in the plant Arabidopsis thaliana. In this talk, I will give a historical introduction to existing data and theories, before presenting our own theory-driven experimental approach, quantitative data and a stochastic model of primary root bending. An iterative process of fitting model to the data will be discussed, and model predictions will be presented.