Gibbs Partitions with Many Components

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In this project we will 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. From today’s viewpoint, such models are among the most 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. In this context it is a fundamental research topic to understand the (global) ‘shape’ of such a partition, when the total size of it becomes large. There are two inherently different settings that must be distinguished: the labeled and the unlabeled case, where in the former the atoms out of which the partition is composed are distinguishable, and in the latter the objects are considered up to symmetry. In the labeled case the model has a neat probabilistic interpretation and consequently the theory is rather well-developed. However, regarding the unlabeled setting, such a connection is not apparent and much less is known, particularly with respect to the distribution (at all scales) of the number of parts in a large partition. The project aims at developing novel methods that will enable us to study systematically such problems in a general setting and improve significantly our understanding of Gibbs partitions.