Combinatorial Stochastic Processes

该项目将继续深入研究大型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（如布朗漂移和Levy过程）之间的联系。要研究的具体课题包括可交换的聚结过程，随机吉布斯分区和碎片，连贯的组合结构，部分交换，一致的系统碎片，连续树的限制，占用问题和随机离散分布的幂律，贝叶斯推理。在过去的几年里，这些主题之间的思想有着丰富的相互作用，预计随着各种组合结构的渐近性模型受到各种应用领域中出现的问题的挑战，还有更多的思想有待发现。一个总的主题是丰富多样的随机离散分布的模型时出现的连续体随机树的叶子中连续分布的质量单位以各种方式分解，如投影到树的分支，或通过切割与泊松点过程的削减沿着树枝的树。这样的建设导致随机离散分布的模型与自然的解释在应用上下文中，例如泊松切割可能代表突变的系统发育树。该项目的结果应该提供更深入的理解模型的随机分区和分区值的过程，特别是破碎和凝聚的过程的演变。这样的结果应该是有价值的，在许多领域，这些过程已经应用过，包括物理学，天文学，遗传学，遗传学，生态学，和文件分析。 特别地，应用具有幂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.