Mathematical Sciences: "Percolation on Trees, Intersections of Random Sets, and Measures of Full Hausdorff Dimension"

佩雷斯 本文主要研究概率论中的几个问题，涉及树索引过程、“小”随机集的交集和Hausdorff维数。 这些问题涉及布朗运动下的“小”集的图像的潜在理论，以及分支过程和某些格子随机游动的范围之间的关系。 布朗样本路径的交集最早是由Dvoretzky、Erdos和Kakutani在40多年前研究的。随机行走路径的交叉点在某些方面更难分析; Lawler（1991）的书介绍了关于它们的大部分已知信息。 提出者对这些问题的方法是由他最近的工作所激发的，他的工作表明某些样本路径（例如布朗运动）与通过分支过程构造的某些“随机康托集”是“等价的”，在这个意义上，相同的确定性集被这两种类型的随机集以正概率击中。 剩下的问题有密切联系的遍历理论，他们涉及Hausdorff维数的措施高尔顿沃森树和不变集的扩大地图。 本文研究了空间随机路径相交的概率论问题。其中一些问题在40年前就被数学家们研究过，但在过去的10年里，数学物理学家们又重新对这个问题产生了兴趣，因为他们已经证明了这个问题与物理学问题的相关性。 分数维的经典概念在本研究中起着重要的作用。一个反复出现的主题是，复杂的配置是最好的理解，找到最均匀的方式来分配他们的质量。 新的数学工具，开发了非常不同的目的，提供了一个令人兴奋的方法来解决这些问题。

Peres The proposer intends to study several problems in Probability theory, concerning tree-indexed processes, intersections of "small" random sets, and Hausdorff dimension. The problems involve potential theory for images of "small" sets under Brownian motion, and the relation between branching processes and ranges of certain lattice random walks. Intersections of Brownian sample paths were first studied by Dvoretzky, Erdos and Kakutani more than forty years ago. Intersections of random walk paths are in some ways harder to analyze; Lawler's (1991) book presents most of what is known about them. The proposer's approach to these problems is motivated by his recent work, which shows that certain sample paths (e.g. of Brownian motion ) are "equivalent" to certain "random Cantor sets" constructed via a branching process, in the sense that the same deterministic sets are hit by these two types of random sets with positive probability. The remaining problems have close ties to ergodic theory; they involve the Hausdorff dimension of measures on Galton Watson trees and on invariant sets for expanding maps. It is proposed to study several problems in probability theory involving intersections of random paths in space. Some of these problems were studied by mathematicians forty years ago, but in the last decade there has been renewed interest, as mathematical physicists have shown the relevance of this topic to questions of physical interest. Classical notions of "Fractional dimension" play an important role in this study. A recurring theme is that complicated configurations are best understood by finding the most uniform way to distribute mass on them. New mathematical tools, developed for very different purposes, offer an exciting approach to these problems.