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

Spatial random permutations (SRP) are a model of statistical mechanics: on a finite set with a notion of distance, a probability measure on permutations of that set is defined in such a way that permutations that map nearby points to each other are preferred. The model has a parameter that determines how strong this preference is. We want to investigate properties of the model in the limit of infinitely large underlying sets. SRP have many connections to other active areas of probability theory and mathematical physics, such as Bose-Einstein condensation, percolation or the Schramm-Löwner evolution. The most important property of SRP is a phase transition: if the preference for mapping points to nearby points is weak, typical spatial permutations will contain cycles with a length that is comparable to the cardinality of the underlying set. Those cycles are absent when the preference is strong. The existence of such a phase transition is known only in very few special models, for most (and for the most relevant) variants of SRP there is only numerical evidence. In the first funding period, DFG funded post-doc Lorenzo Taggi and myself developed a novel and quite promising tool for investigating SRP. It is a special coupling method, making use of a spatial Markov property of the model, and allowing in particular to prove for the first time exponential decay of correlations in certain parameter regimes. Decay of correlations is a basic and very important property in many systems of statistical mechanics, and has before been inaccessible for SRP since particular long ranged dependencies prevented the use of classical proof recipes. In the relevant paper, the coupling method has been used to investigate the geometry of forced long cycles in regimes where long cycles are naturally absent. The scope of the method is much wider, though. The aim of the research which we ask to be funded by this application is to systematically investigate, extend and apply the method in the context of SRP. In particular, it seems to be possible to prove exponential decay of special correlations in regimes where long cycles are thought to be present. Using certain modifications of the method, even a proof of the existence of long cycles seems possible. The latter is one of the most important goals in the research about SRP. Because of the great promise of the coupling method, the aims formulated in the original application that have not yet be achieved will be postponed. However, they still constitute very attractive alternative research directions that can be re-activated as needed.

• DFG Programme: Research Grants