Topics in Discrete Probability and Stochastic Processes

大型离散随机过程的行为有时可以用极限随机连续空间过程来研究。这种“弱收敛”方法在几个领域得到应用。随机聚结处理的是粒子按照一定的速率核合并成越来越大的团簇。加性速率核的特殊情况正在被详细研究，它具有与连续统随机树相关的丰富结构。还在研究一种新的过程，它结合了结合、迁移和清除，似乎具有有趣的相变。在不同的方向上，人们对重建系统发育树(表明生物物种之间的关系)的马尔可夫链蒙特卡罗方法越来越感兴趣。利用弱收敛方法研究了树空间上目标分布抽样的玩具模型。研究的一般领域是行为具有随机性的大系统。随着这样的系统变得更大，它们的行为变得更难分析，无论是通过数学理论还是通过计算机模拟。但在某些情况下，可以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来克服这一困难。这一方法正在两个领域得到应用。其中一个领域是改进质量随机聚集模型的分析，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、云中雨滴的凝聚，以及早期宇宙中星系的形成)。另一个领域是重建生物物种之间关系的算法的性能分析，特别是性能如何随物种数量的变化而变化。

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.