Marielle Simon (Université Lyon 1)

Title: On exclusion processes with phase separation

Abstract:

"Stochastic lattice gases" are models of interacting particles subject to stochastic dynamics. They have been widely studied for about thirty years by both mathematicians and physicists. Their structure makes it possible to analyse them rigorously, while illustrating numerous physical phenomena: in particular, one of the main objectives is to prove rigorously the convergence of the microscopic system towards a macroscopic PDE, after rescaling in time and space (also known as the ‘hydrodynamic limit’).

In the first lecture I will give several illustrations of this convergence result thanks to the most well-known example, namely the symmetric simple exclusion process. In the second lecture, I will show how to enrich these models in order to derive some phase separation at the macroscopic level, with a free boundary that moves within the system.

Michel Mandjes (Leiden University)

Title: Dynamic random graphs: analysis and inference

Abstract:

The bulk of the random graph literature concerns models that are of an inherently static nature, in that features of the random graph at a single point in time are considered. There are strong practical motivations, however, to consider random graphs that are stochastically evolving, so as to model networks’ inherent dynamics. 

In this talk I’ll discuss a set of dynamic random graph mechanisms and their probabilistic properties. Key results cover functional diffusion limits for subgraph counts (describing the behaviour around the mean) and a sample-path large-deviation principle (describing the rare-event behaviour, thus extending the seminal result for the static case developed by Chatterjee and Varadhan).

The last part of my talk will be about estimation of the model parameters from partial information. We for instance demonstrate how the model’s underlying parameters can be estimated from just snapshots of the number of edges. We also consider settings in which particles move around on a dynamically evolving random graph, and in which the graph dynamics are inferred from the movements of the particles (i.e., not observing the graph process).