Combinatorial Stochastic Processes

该项目将继续研究大有限集上的组合结构理论（如树、森林、映射和分区）与随机过程理论（如布朗偏移和利维过程）之间的深层联系。要研究的具体主题包括可交换的凝聚过程、随机吉布斯分区和碎片、连贯组合结构、部分可交换性、碎片的一致系统、连续体树极限、随机离散分布的占用问题和幂律，以及贝叶斯推理。在过去的几年里，这些学科之间的思想有了丰富的相互作用，随着各种组合结构的渐近性的现有模型受到各种应用领域中出现的问题的挑战，预计还会有更多的东西有待发现。一个普遍的主题是随机离散分布的丰富多样的模型，当连续分布在连续随机树的叶子中的质量单位以各种方式分解时，例如通过投影到树的分支上，或通过泊松点过程沿着树的分支切割。这样的构造导致随机离散分布的模型在应用环境中具有自然的解释，例如泊松切割可能代表系统发育树中的突变。该项目的结果应提供对随机分区和分区值过程的演化模型的更深入理解，特别是破碎和凝聚过程。这样的结果在许多领域应该是有价值的，这些领域已经应用了这样的过程，包括物理学、天文学、遗传学、系统发育、生态学和文献分析。特别是，应用具有幂律尾的随机离散分布来模拟科学文献中主题的分布，可以为大量科学文献的分类和导航提供改进的方法。

The project will continue an ongoing study of deep connections between the theory of combinatorial structures on a large finite set, such as trees, forests, mappings, and partitions, and the theory of stochastic processes such as Brownian excursion and Levy processes. Specific topics to be studied include exchangeable coalescent processes, random Gibbs partitions and fragmentations, coherent combinatorial structures, partial exchangeability, consistent systems of fragmentations, continuum tree limits, occupancy problems and power laws for random discrete distributions, and Bayesian inference. There has been a rich interplay of ideas between these subjects in the last few years, and it is expected that much more remains to be discovered as present models for the asymptotics of various combinatorial structures are challenged by problems arising in various application areas. One general theme is the rich variety of models of random discrete distributions which arise when unit of mass continuously distributed in the leaves of a continuum random tree is decomposed in various ways, such as by projection onto a branch of the tree, or by cutting with a Poisson point process of cuts along branches of the tree. Such constructions lead to models for random discrete distributions with natural interpretations in application contexts, for instance the Poisson cuts may represent mutations in a phylogenetic tree.Results of the project should provide deeper understanding of models for the evolution of random partitions and partition-valued processes, particularly processes of fragmentation and coagulation. Such results should be of value in the numerous fields where such processes have been applied before, including physics, astronomy, genetics, phylogeny, ecology, and document analysis. In particular, the application of random discrete distributions with power law tails to model the distribution of topics among scientific documents may provide improved methods for classification and navigation of large bodies of scientific literature.