Mathematical Sciences: Random Trees and Tree-Indexed Processes

树索引过程是一种统一而有力的研究经典树相关概率模型的方法。这些模型的例子有：分支随机漫步，分支马尔可夫链，树上的第一通道渗透，并行搜索，随机和非随机康托集以及双曲流形。目前关于树索引过程的工作主要集中在扩展随机集的Hausdorff维的一些结果，以及确定给定树索引过程的范围空间子集为极值的精确容量准则。具有分支结构或树形结构的概率模型出现在许多科学领域。许多关于分支过程的经典工作都是由群体遗传学中出现的随机家谱所激发的；级联理论包括原子链式反应和降雨模型；最近，随机树被用于研究空间模型中的自相似性（分形行为）。通过将这些模型的公共结构抽象为“树索引过程”，我们已经能够证明关于这类模型的许多新结果，并加强了先前已知的结果。

Tree-indexed processes are a way of studying classical tree-related probability models in a unified and more powerful way. Examples of such models are: branching random walk, branching Markov chains, first passage percolation on a tree, parallel searches, random and non-random Cantor sets and hyperbolic manifolds. Current work on tree-indexed processes is focused on extending some results on Hausdorff dimensions of random sets and on determining exact capacity criteria for subsets of the range space to be polar for a given tree-indexed process. Probability models with branching or tree structure arise in many areas of science. Much of the classical work on branching processes was motivated by random family trees that arise in population genetics; the theory of cascades encompasses atomic chain reactions and rainfall models; more recently, random trees have been used to study self-similarity (fractal behavior) in spatial models. By abstracting the common structure of these models into "tree-indexed processes", we have been able to prove many new results about this class of models, as well as strengthening previously known results.