Gibbssche Partitionen mit vielen Komponenten
Theoretische Informatik
Zusammenfassung der Projektergebnisse
The aim of this project was to study models of partitions of sets, where each individual part and possibly the partition as a whole are equipped with weights. Such models are rather well-known under the name Gibbs Partitions, and they appear naturally in a large number of areas, including Probability, Combinatorics and Statistical Physics. These models are important and fundamental, as from today’s viewpoint they provide the very general means of composing complex structures out of simpler ones, and they have a prominent place in modern theories of asymptotic enumeration and applied Probability. The aim of this project was on describing and understanding the (global) ‘shape’ of such partitions, when the total size becomes large. Moreover, the focus was on the unlabeled case, where the objects are considered up to symmetry. This model is particularly interesting and challenging, as at the time the proposal was written there was no appropriate description of such classes that would enable us to perform an analysis of typical structural properties. Within the project we developed an appropriate description of Gibbs Partitions in the unlabeled setting that is based on recent developments in the context of Polya-Boltzmann samplers for combinatorial structures. This allows us to study algorithms that generate objects according to the desired distribution, which is a much simpler problem: the algorithms make plently of independent choices, and the generated objects depends ‘smoothly’ on that choices, so that studying the objects amount to studying properties of independent (but not identically distributed) random variables. This reduction allowed us to describe and enumerate unlabeled set partitions in great detail, to develop new sampling algorithms, and to discover a simple limit theory for such objects.
Projektbezogene Publikationen (Auswahl)
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Asymptotic enumeration and limit laws for multisets: The subexponential case. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 60(1).
Panagiotou, Konstantinos & Ramzews, Leon
