Trimmed trees and embedded particle systems

Trimming a Tree and the Two-Sided Skorohod Reflection

The $h$-trimming of a tree is a natural regularization procedure which consists in pruning the small branches of a tree: given $h\geq0$, it is obtained by only keeping the vertices having at least one leaf above them at a distance greater or equal to $h$. The $h$-cut of a function $f$ is the function of minimal total variation uniformly approximating the increments of $f$ with accuracy $h$, and can be explicitly constructed via the two-sided Skorohod reflection of $f$ on the interval $[0,h]$. In this work, we show that the contour path of the $h$-trimming of a rooted real tree is given by the …

嵌入在$\mathbb{R}^{\mathfrak{D}}$中的随机Yule树的标度性质

Scaling properties of a random Yule tree embedded in $\mathbb{R}^{\mathfrak{D}}$

本研究针对Ajazi等人提出的神经元网络模型中的随机Yule树，通过推导其分布的标度关系，用基础技术得到时间趋于无穷时树几何性质的渐近结果。

The density of the ISE and local limit laws for embedded trees

It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) Hölder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by Bousquet-Mélou on convergence of the vertical profile at a …

Erdős-Rényi 图拉普拉斯算子的谱隙与嵌入树

Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph

本文研究了 Erdős-Rényi 图在特定平均度区间内拉普拉斯算子的谱隙，证明其最小非零特征值以高概率由仅通过一条边连接巨连通分量的规模为 $t$ 的线状小子图决定，并给出了精确的渐近表达式。

树嵌入与站点渗透中的非唯一性

Tree embeddings and nonuniqueness in site percolation

论文证明了在适当嵌入的平面图上伯努利站点渗透的非唯一性定理，并提出了超越平面性的通用连通性原则。研究发现，对于最小度数至少为7的无限连通图G，临界概率p_c^site(G)小于1/2，并在特定概率范围内存在无限多无限开放簇。

A class of multifractal processes constructed using an embedded branching process

We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton-Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at st…