Aims: A complex system undergoes condensation if a positive fraction of an observed quantity concentrates in a single state, asymptotically as time (or some other model parameter) goes to infinity. Condensation is a characteristic feature of many complex systems including wealth condensation in macroeconomic systems, gelation of molecules, the physical phenomenon of Bose-Einstein condensation, and the formation of traffic jams. In this project we investigate how condensation can build up, or emerge, in complex systems. This will be done by investigating relatively simple mathematical toy models. In these models we zoom into a small neighbourhood of the condensation state at a finite time and rescale the distribution of the observable around this site. If the scaled distribution converges as time goes to infinity and scale goes to zero we call the limit a condensing wave. In other words, the condensing wave describes the shape of the distribution of the observable before it collapses to form the condensate. A recent pilot study by the investigator and his collaborators has shown that for several models from very different contexts the shape of the condensing wave is universal, i.e., it does not depend on the specific model details. In all the examples the shape was that of a well-known probability distribution, the gamma distribution. The pressing questions resulting from these observations are:

• How large is the universality class of complex systems exhibiting condensation in this form?
• Are there other universality classes with different wave shapes?
• How can we characterize systems in different universality classes in terms of accessible parameters?
• What features other than the shape of condensing wave are characteristic of a given class?

The project will address these questions for a variety of models of different complexity.

Principal investigator: Peter Mörters
Postdoctoral Researcher: Cécile Mailler

Collaborators:

• Volker Betz (Darmstadt)
  
• Steffen Dereich (Münster)
  
• Daniel Ueltschi (Warwick)
  
  

• Prob-L@B Workshop "Condensation phenomena in stochastic systems" Bath 4.7.-6.7.2016.

• Emergence of condensation in Kingman's model of selection and mutation
  Steffen Dereich and Peter Mörters. Acta Applicandae Mathematicae. 127 (2013) 17-26.
  
• Cycle length distributions in random permutations with diverging cycle weights
  Steffen Dereich and Peter Mörters. Random Structures and Algorithms. 46 (2015) 635-650.
  
• Condensation and symmetry-breaking in the zero-range process with weak site disorder
  Cécile Mailler, Peter Mörters and Daniel Ueltschi. Stochastic Processes and their Applications. 126 (2016) 3283-3309.
  
• Non-extensive condensation in reinforced branching processes
  Steffen Dereich, Cécile Mailler and Peter Mörters. To appear in Annals of Applied Probability.
  
• On the shape of condensing waves
  Volker Betz, Steffen Dereich, Peter Mörters and Daniel Ueltschi. In preparation.
  

• Condensation in models of selection and mutation
  Random Combinatorial Structures and Statistical Mechanics, Venice, May 2013.
• The universal shape of Bose-Einstein condensates
  Combinatorics and Statistical Mechanics, Warwick, April 2014.
• Condensation in the non-homogeneous zero-range process
  Probability Workshop, Oxford, May 2014.
• Emergence of condensation in a self-organised growth model
  Workshop Interplay of Analysis and Probability in Applied Mathematics, Oberwolfach, July 2015.

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