Gibbs Partitions with Many Components

具有多个组件的吉布斯分区

• 批准号: 411724275
• 负责人: Professor Dr. Konstantinos Panagiotou
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/411724275?language=en
• 关键词: Gibbs; Partitions; Many; Components

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.

在这个项目中，我们将研究集合划分的模型，其中每个单独的部分以及可能的整个划分都配备了权重。此类模型以吉布斯分区的名称而闻名，它们自然出现在许多领域，包括概率、组合学和统计物理学。从今天的角度来看，此类模型是从简单结构组成复杂结构的最通用方法之一，并且它们在现代渐近枚举和应用概率理论中占有重要地位。在这种情况下，当分区的总大小变大时，了解这种分区的（全局）“形状”是一个基础研究课题。必须区分两种本质上不同的设置：标记和未标记的情况，其中在前者中组成分区的原子是可区分的，在后者中对象被认为是对称的。在标记的情况下，模型具有简洁的概率解释，因此该理论相当完善。然而，对于未标记的设置，这种联系并不明显，而且知之甚少，特别是关于大分区中部件数量的分布（在所有尺度上）。该项目旨在开发新颖的方法，使我们能够在一般环境下系统地研究此类问题，并显着提高我们对吉布斯分区的理解。