Topics in Discrete Probability and Stochastic Processes

9970901大型离散随机过程的行为有时可以用有限随机连续空间过程来研究。 这种"弱趋同”方法适用于若干领域。 随机聚结是指粒子按照一定的速率核合并成越来越大的团簇。 特殊情况下的添加剂率内核，它有一个丰富的结构有关的连续随机树，正在详细研究。 一个新的过程相结合的移民和迁移，似乎有一个有趣的相变也正在研究。 在不同的方向，有越来越多的兴趣在马尔可夫链蒙特卡罗方法重建系统发育树（指示生物物种之间的关系）。 利用弱收敛方法研究了树空间上目标分布抽样的一些玩具模型。一般研究领域涉及行为具有随机性的大系统。 随着这些系统变得越来越大，它们的行为变得越来越难以通过数学理论或计算机模拟来分析。 但在某些情况下，通过将足够大的有限系统的行为与某个固定的无限大系统的行为联系起来，可以克服这个困难。 这一方法正应用于两个领域。一个领域是改进质量随机聚类模型的分析，这是许多科学领域感兴趣的主题（例如物理化学中的聚合物，云中雨滴的凝结，以及早期宇宙中星系的形成）。 另一个领域是分析用于重建生物物种之间关系的算法的性能，特别是在性能如何与物种数量成比例方面。

9970901The behavior of large discrete random processes can sometimes be studied in terms of a limiting random continuous-space process. This ``weak convergence" methodology is applied in several areas. Stochastic coalescence deals with the merging of particles into larger and larger clusters according to some rate kernel. The particular case of the additive rate kernel, which has a rich structure relating to continuum random trees, is being investigated in detail. A novel process combining coalescence with immigration and removal and which appears to have an interesting phase transition is also being studied. In a different direction, there is increasing interest in Markov chain Monte Carlo methods for reconstructing phylogenetic trees (indicating relationships between biological species). Some toy models for sampling from target distributions on tree-space are studied via the weak convergence methodology.The general area of research concerns large systems with random aspects to their behavior. As such systems become larger, their behavior gets more difficult to analyze, either via mathematical theory or via computer simulation. But in some cases it is possible to overcome this difficulty by relating the behavior of sufficiently large finite systems to that of some fixed infinite-size system. This methodology is being applied in two areas. One area is refining analysis of models of random clustering of mass, a topic of interest in many areas of science (e.g. polymers in physical chemistry, condensation of raindrops in clouds, and formation of galaxies in the early universe). The other area is the analysis of performance of algorithms for reconstructing relationships between biological species, specifically in how performance scales with number of species.