Gibbs Partitions with Many Components

在这个项目中，我们将研究集合的划分模型，其中每个单独的部分和可能的划分作为一个整体都配备了权重。这样的模型以吉布斯分区的名义闻名于世，它们自然地出现在许多领域，包括概率论、组合学和统计物理学。从今天的观点来看，这些模型是用简单的结构组成复杂结构的最普遍的方法之一，它们在现代渐近枚举理论和应用概率论中占有突出地位。在这种情况下，当分区的总规模变大时，了解分区的（整体）“形状”是一个基本的研究课题。有两种本质上不同的情况必须加以区分：标记的情况和未标记的情况，在前者中，组成分割的原子是可区分的，在后者中，对象被认为是对称的。在标记的情况下，模型有一个简洁的概率解释，因此理论相当发达。然而，对于未标记的设置，这种联系并不明显，而且所知道的也少得多，特别是关于一个大分区中零件数量的分布（在所有尺度上）。该项目旨在开发新的方法，使我们能够在一般情况下系统地研究这些问题，并显著提高我们对吉布斯分区的理解。

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.