Mathematical Sciences: Some Problems in Probability Theory

9625458 Kesten摘要 研究者研究概率论中的几个问题。更具体地说，主要的问题涉及第一次通过渗流，普通渗流和优化泛函的中心极限定理。对于通常的第一次通过渗流模型的提议者希望建立一些一般性质的渐近形状的一组点，可以达到在一个大的时间;这个渐近形状应该严格包含在一个romboid和立方体之间。他还在研究当允许顶点的通过时间取决于已经到达的邻居数量时，渐进形状如何变化。在普通渗流中，研究者（与几位同事）正在研究一个大盒子中最大簇的大小分布，并希望在一个完整的图（Erdos-Renyi随机图）上为最大簇建立一些类似的已知结果，其中每个边都以一定的概率存在。最后，在一个不同的领域，他正试图证明中央极限定理的一些优化功能有点类似于最小生成树。 这个建议中的问题部分是由统计物理和某些优化算法的行为引起的。通过这项工作的第一部分，人们可以深入了解一些随机增长模型的大时间行为，以及局部规则（即，小邻域中系统行为的规则）影响系统的全局行为。优化算法的研究涉及依赖于大量随机选择的点之间的距离的量。 这些研究有望使我们了解一些困难和重要的优化问题的典型行为，目前还没有好的计算方法。

9625458 Kesten ABSTRACT The investigator studies several problems in probability theory. More specifically, the main problems deal with first-passage percolation, ordinary percolation and central limit theorems for optimizing functionals. For the usual first-passage percolation model the proposer wishes to establish some general properties of the asymptotic shape of the set of points which can be reached in a large time; this asymptotic shape should be strictly contained between a romboid and a cube. He also is investigating how the asymptotic shape varies when one allows the passage-time of a vertex to be dependent on how many neighbors have already been reached. In ordinary percolation the investigator (with several colleagues) is investigating the distribution of the size of the largest cluster in a big box, and hopes to establish a number of analogues to known results for the largest cluster on a complete graph in which each edge is present with a certain probability (the Erdos-Renyi random graph). Finally, in a different area, he is attempting to prove central limit theorems for some optimizing functionals which are somewhat similar to minimal spanning trees. The problems in this proposal are motivated in part by statistical physics and by behavior of certain optimization algorithms. Through the first part of this work one gains insight into the large time behavior of some stochastic growth models, and how variations in the local rules (i.e., rules for the behavior of the system in small neighborhoods) influence the global behavior of the system. The research on optimization algorithms deal with quantities which depend on the distances between a large number of randomly chosen points. These studies are expected to give us an understanding of the typical behavior of some difficult and important optimization problems for which no good computational methods are known at present.