Mathematical Sciences: Random Trees and Tree-Indexed Processes

树索引进程是研究经典 树相关的概率模型在一个统一的，更强大的 路上了 这种模型的例子有：分支随机游走， 分支马尔可夫链，树上的第一次通过渗流， 并行搜索，随机和非随机康托集， 双曲流形 目前关于树索引进程的工作是 重点推广了Hausdorff维数的一些结果， 随机集和确定准确的能力标准， 对于给定的树索引，范围空间的子集是极坐标的 过程 具有分支或树结构的概率模型出现在 许多科学领域。 许多关于分支的经典著作 过程的动机是随机的家谱， 群体遗传学;级联理论包括原子 连锁反应和降雨模型;最近，随机树 已经被用来研究自相似性（分形行为）， 空间模型 通过抽象这些的共同结构， 模型转化为“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，以及 加强了以前已知的成果。

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.