Long cycles of spatial random permutations: Size, geometry and dynamics

• Applicant: Professor Dr. Volker Betz
• Subject Area: Mathematics
• Term: from 2014 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255533188

• Final Report Year: 2019

Final Report Abstract

Spatial random permutations (SPR) are a model that is simple to describe in plain words, but difficult to analyse. A special version of SRP is as follows: imagine a cube-shaped subset of Zd with side length L and consider the set of all permutations on it with the property that each point in ΛL is either mapped to itself or to one of its neighbours. Let n(π) be the number of fixed points (points that are mapped to themselves) under a permutation π. The model consists in choosing a random permutation from the set of all possible permutations, and the law of this randomness should only depend on n(π). More precisely, the probability of choosing a certain permutation π should be proportional to eαn(π) . Thus, the parameter α regulates how likely permutations with many fixed points are. The interest in such models comes mainly from their connection with the theory of Bose-Einstein condensation. In this context, the important question is the existence of long cycles: when L is very large, is it true or not that with reasonably high probability, a randomly sampled permutation will have cycles that contain a fraction (e.g. more than 1/2, or more than 1/100) of the available points)? Does this answer depend on the parameter α? This is a very difficult question to answer rigorously, although there is overwhelming numerical evidence for the existence of some α0 such that for α > α0 , no long cycles exist, while for α < α0 , long cycles exist. In this project, we made the first steps towards solving this question in a special case. One of the most important outcomes is a paper where, using a method (’reflection positivity’) that had to be adapted to the current situation, it was shown that SRP indeed have long cycles in the (degenerate) case α = −∞, i.e. in situations where we do not allow fixed points. This result may seem quite special, but it is an important first step towards understanding more general models of SRP.

Publications

• The number of cycles in random permutations without long cycles is asymptotically Gaussian, ALEA 14, 427–444 (2017)
  V. Betz and H. Schäfer
  
• Critical density of activated random walks on Zd and general graphs. Annals of Probability, Vol 46, No 4, 2190-2220 (2018)
  A. Stauffer and L. Taggi
  
• Shifted critical threshold for the loop O(n) model with arbitrarily small n, Electronic Communications in Probability, Vol 23, No 96, Pag 1-9 (2018)
  Taggi, Lorenzo
  
• Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations, Electron. J. Probab. 24 (2019), paper No. 74
  Betz, Volker & Taggi, Lorenzo
  
• Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N
  Lees, Benjamin & Taggi, Lorenzo
  
• Uniformly positive correlations in the dimer model and phase transition in lattice permutations on Zd, d > 2, via reflection positivity
  L. Taggi
  
• (2020) Random permutations without macroscopic cycles. Ann. Appl. Probab. (The Annals of Applied Probability) 30 (3) 1484-1505
  Betz, Volker; Schäfer, Helge & Zeindler, Dirk
  
• Interacting self-avoiding polygons, Ann. Inst. H. Poincare (B), Probability and Statistics (2019)
  Betz, Volker; Schäfer, Helge & Taggi, Lorenzo