Combinatorial Stochastic Processes

该项目将继续正在进行的关于大型有限集上的组合结构理论(如树、森林、映射和划分)与随机过程理论(如布朗游程和Levy过程)之间的深层联系的研究。要研究的具体课题包括可交换的合并过程、随机Gibbs划分和碎片、一致的组合结构、部分可交换性、一致的碎片系统、连续统树限制、随机离散分布的占用问题和幂定律，以及贝叶斯推理。在过去的几年里，这些学科之间存在着丰富的思想相互作用，随着各种组合结构的现有模型受到各种应用领域出现的问题的挑战，预计还会有更多的发现。一个总的主题是当连续分布在连续统随机树的树叶中的质量单位以各种方式分解时产生的丰富的随机离散分布模型，例如通过投影到树的分支上，或者通过沿着树的分支切割的泊松点切割过程。这样的构造导致了在应用环境中具有自然解释的随机离散分布的模型，例如泊松割可能代表系统发生树中的突变。该项目的结果应该为随机划分和分割值过程的演化模型提供更深入的理解，特别是碎片和凝聚过程。这样的结果应该对以前应用过这种方法的许多领域有价值，包括物理学、天文学、遗传学、系统发育学、生态学和文件分析。特别是，应用具有幂定律尾部的随机离散分布来对科学文献中的主题分布进行建模，可能会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.