Research Group of Professor Mörters

Many of the phenomena in the complex world in which we live have a rough description as a large network of interacting components. Random network theory tries to describe the global structure of such networks from basic local principles using stochastic models. Models we are interested in include preferential attachment networks, spatially embedded networks and time-dependent inhomogeneous random graphs.

• Random networks with sublinear preferential attachment: The giant component
  Steffen Dereich and Peter Mörters. Annals of Probability, 41 (2013) 329-384.
• Spatial preferential attachment networks: Power laws and clustering coefficients
  Emmanuel Jacob and Peter Mörters. Annals of Applied Probability, 25 (2015) 632-662.

It is a crucial challenge in probability to understand the emergent behaviour of systems of particles interacting with each other or with an external environment when the system size is increasing. Research in our group is using large deviation theory as well as multiscale analysis to understand the limiting behaviour of a range of models, including infections spreading on particles migrating in space.

• Multi-scale Lipschitz percolation of increasing events for Poisson random walks
  Peter Gracar and Alexandre Stauffer. Annals of Applied Probability, 29 (2019) 376-433.
• The Semi-Infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products
  Horacio González Duhart, Peter Mörters and Johannes Zimmer. Potential Analysis, 48 (2018) 301-323.

We are interested in the problem how an inhomogeneous environment can influence the behaviour of a stochastic process taking place in such an environment. Examples studied in our group include diffusion process in an irregular random potential (parabolic Anderson model) and infection processes on inhomogeneous random networks. The general phenomena observed in this context include ageing, intermittency and metastability.

• The universality classes in the parabolic Anderson model
  Remco van der Hofstad, Wolfgang König and Peter Mörters. Communications in Mathematical Physics, 267 (2006) 307-353.
• The contact process on scale-free networks evolving by vertex updating
  Emmanuel Jacob and Peter Mörters. Royal Society Open Science, 4 (2017) Issue 5.

Condensation occurs in a system if a proportion of a distributed observable quantity concentrates asymptotically in a single location. The physical phenomenon of Bose-Einstein condensation (where the observed quantity is energy and a positive fraction of bosons occupy the lowest quantum state) is just one example. We are interested in the dynamics of the condensation process for a range of models exhibiting condensation, including stochastic processes with reinforcement, the zero-range process, and non-stochastic (PDE based) models.

• Non-extensive condensation in reinforced branching processes
  Steffen Dereich, Cécile Mailler and Peter Mörters. Annals of Applied Probability, 27 (2017), 2539-2568.
• The shape of the emerging condensate in effective models of condensation
  Volker Betz, Steffen Dereich and Peter Mörters. Annales Henri Poincaré, 19 (2018) 1869-1889.