Mathematical Sciences: Some Problems in Probability Theory

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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.