The total path length of split trees

The asymptotic distribution of cluster sizes for supercritical percolation on random split trees

We consider the model of random trees introduced by Devroye (1999), the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. We also show that the approach developed in this work may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we study also the case of $d$-regular trees.

The fluctuations of the giant cluster for percolation on random split trees

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant clust…

Random hyperplane search trees in high dimensions

Given a set S of n \geq d points in general position in R^d, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits. We investigate the structural distributions of such random trees with a particular focus on the growth with d. A blessing of dimensionality arises--as d increases, random hyperplane splits more closely resemble perfectly balanced splits; in turn, random hyperplan…

临界Beta分裂随机树III：可交换划分表示与边缘树

The Critical Beta-splitting Random Tree III: The exchangeable partition representation and the fringe tree

本文研究临界beta分裂随机树的渐近性质，提出规范嵌入连续时间模型与显式刻画随机叶子对应的极限边缘分布两种概率方法，为该模型的结构分析提供新工具。

On the asymptotic internal path length and the asymptotic Wiener index of random split trees

The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.

Self-similar fragmentations derived from the stable tree II: splitting at nodes

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Levy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively …

Voronoi cells in random split trees

We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in cont…

Tree limits and limits of random trees

We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton-Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

Open and increasing paths on N-ary trees with different fitness values

Consider a rooted N-ary tree. For every vertex of this tree, we atttach an i.i.d. Bernoulli random variable. A path is called open if all the random variables that are assigned on the path are 1. We consider limiting behaviors for the number of open paths from the root to leaves and the longest open path. In addition, when all fitness values are i.i.d. continuous random variables, some asymptotic properties of the longest increasing path are proved.

A tree approach to $p$-variation and to integration

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, Lévy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is…

临界贝塔分裂随机树：高度及相关结果

The Critical Beta-splitting Random Tree: Heights and Related Results

本文研究临界贝塔分裂随机树的连续时间模型，精确求出随机叶子时间高度和边高度的前两阶矩并证明中心极限定理，同时分析叶子高度相关性、分裂次数期望、树高度尾部界及随机子树大小的极限分布。

Some notes on trees and paths

These notes cover background material on trees which are used in the paper `On uniqueness of the signature of a path of variation and the reduced path group'.

通过最小最大决策树稳定分裂

Stabilizing the Splits through Minimax Decision Trees

本文提出MinimaxSplit决策树，通过最小化最坏情况下的子节点风险来替代CART，适用于回归和分类任务，并在结构异质数据上表现更优。

临界 Beta 分裂随机树 IV：叶高的 Mellin 分析

The Critical Beta-splitting Random Tree IV: Mellin analysis of Leaf Height

本文通过极限树结构与 Mellin 变换分析，对临界 Beta 分裂随机树中叶高的期望、方差、正态近似及大偏差行为给出更精确的估计。