Jean-François Marckert, Grégory Miermont
We prove that a uniform, rooted unordered binary tree with $n$ vertices has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Consider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton-Watson tree with finite variance and conditioned to have size n, converges as $n\to\infty$ to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov-Prokhorov topology. This yields a multi-dimensional extension of a limit theor…
We embed Duquesne and Le Gall's stable tree into a binary compact continuum random tree (CRT) in a way that solves an open problem posed by Goldschmidt and Haas. This CRT can be obtained by applying a recursive construction method of compact CRTs as presented in earlier work to a specific distribution of a random string of beads, i.e. a random interval equipped with a random discrete measure. We also express this CRT as a tree built by replacing all branch points of a stable tree by rescaled i.i.d. copies of a Ford CRT. Some of these developments are carried out in a space of infinity-marked m…
We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $ψ$ be a critical branching mechanism. We set $ψ_θ(\cdot)=ψ(\cdot+θ)-ψ(θ)$. Let $Θ=(θ_\infty,+\infty)$ or $Θ=[θ_\infty,+\infty)$ be the set of values of $θ$ for which $ψ_θ$ is a branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $(\ct_θ,θ\inΘ)$, such that $\mathcal{T}_θ$ has …
We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the Lévy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $λ$, see Duquesne and Le Gall (2002). The tree-valued process, as $λ$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $θ$ and the intensity $λ$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forwar…
Convergence to the Brownian CRT for critical branching Markov processe
证明有限方差临界分支过程在重标度下族谱树收敛到布朗连续随机树，建立普适标度极限。
We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that "macroscopic" splitting events are rare, we show that Markov branching trees admit the…
On the cover time of Brownian motion on the Brownian continuum random tree
本文证明了布朗连续随机树上的布朗运动的覆盖时间等于局部时间处处严格为正的时刻的下确界，并利用递归自相似性和新的Ray-Knight定理，推导出临界有限方差Galton-Watson树上简单随机游走的覆盖时间在退火律下依分布收敛到该覆盖时间。
Rémy's diffusion on Brownian trees
研究Rémy算法在连续树空间上的推广，构造布朗连续随机树上的局部扩散过程，并证明其遍历性。
We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galt…
The Aldous diffusion: a stationary evolution of the Brownian CRT
本文构造了Aldous猜想的连续树扩散过程，其分支点质量按Wright-Fisher扩散演化，证明其平稳分布为布朗CRT，并借此解决了Aldous关于分支图马尔可夫链标度极限的猜想。
We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform $p$-angulations. As their number of vertices $n$ goes to infinity, we show that these random graphs, rescaled by $n^{-1/2}$, converge in the Gromov--Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geode…
Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent
本文构造了加性凝聚的 Aldous-Pitman 与 Bertoin 两种表示的耦合，利用布朗 CRT 切割树的探索算法，统一了两种看似不同的构造。
The $γ$-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $γ$-stable domain, $γ\in (1, 2]$. They form a specific class of Lévy trees (introduced by Le Gall and Le Jan in1998) and the Brownian case $γ= 2$ corresponds to Aldous Continuum Random Tree (CRT). In this paper, we study fine properties of the mass measure, that is the natural measure on $γ$-stable trees. We first discuss the minimum of the mass measure of balls with radius $r$ and we show that this quantity is of order $r^{\fracγ{γ-1}} (…
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $γ$-Liouville quantum gravity (LQG) for $γ\in (0,2)$ if we take the correlation to be $-\cos(πγ^2/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the q…
We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale.
Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees
本文研究条件临界稳定 Galton–Watson 树随机删边所得碎裂过程，证明其重标度后收敛到 $\alpha$-稳定 Lévy 树的比例切割碎裂，并通过稳定 Lévy 游程划分给出构造，推广 Brownian CRT 结果。
Growing Self-Similar Markov Trees
本文研究了自相似马尔可夫树中通过剪枝分支获得质量为1/2的布朗树的机制，揭示了其与叶生长算法缩放极限的联系。
The circle packing and Riemann uniformization embedding of the mated-CRT map converges to Liouville quantum gravity
本文证明 mated-CRT 映射在圆填充和黎曼均匀化嵌入下收敛于刘维尔量子引力，并推广了 He-Schramm 的收敛结果，为后续生成树加权平面映射的收敛奠定基础。
We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the uniform spanning trees of the Erdös-Rényi sequence $(G(n,p))_{n \geq 1}$ for any fixed $p \in (0,1]$, and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law o…
Invariance principles for random walks in random environment on trees
本文利用电阻度量测度空间的泛函极限定理，研究树上随机环境随机游走的标度极限，证明有偏游走与边增强游走分别收敛为 CRT 上随机 Gaussian 势中的 Brown 运动。
Self-similar Markov trees and scaling limits
该研究聚焦自相似马尔可夫树，结合实树递归构造与Lamperti观察构建，通过长度及调和测度实现显式脊柱分解，为多种随机实树提供框架并建立Galton-Watson树不变性原理。
We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the sc…
The scaling limit of random 2-connected series-parallel maps
本文研究加权随机2-连通串并联地图的标度极限，证明在可积性条件下，距离缩放$n^{-1/2}$后收敛到Aldous连续随机树的常数倍，证明基于树与地图间的双射及马尔可夫链论证。
We study the scaling limits of looptrees associated with Bienaymé--Galton--Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a "cycle" whose size is its degree. First, we consider BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution. We prove that the Brownian CRT is the scaling limit of the associated looptrees, thereby confirming a prediction of [CK14b]. Then, we deal with BGW trees whose offspring distribution is critical and heavy-tailed. We show that the scaling limit of the associated looptrees is a mul…
Let $(\mathcal{T},d)$ be the random real tree with root $ρ$ coded by a Brownian excursion. So $(\mathcal{T},d)$ is (up to normalisation) Aldous CRT \cite{AldousI} (see Le Gall \cite{LG91}). The $a$-level set of $\mathcal{T}$ is the set $\mathcal{T}(a)$ of all points in $\mathcal{T}$ that are at distance $a$ from the root. We know from Duquesne and Le Gall \cite{DuLG06} that for any fixed $a\in (0, \infty)$, the measure $\ell^a$ that is induced on $\mathcal{T}(a)$ by the local time at $a$ of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in $\mathcal…
Recursive Paintboxes and the Martin Boundary of the Hoffman Rooted-Tree Graph
本文确定了Hoffman有根树图的Martin边界，证明其由递归油漆盒参数化，并给出了递归Ewens族的极值分解。
We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural a…
We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is a least the sum of the sizes of its children. Assuming that macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased chil…
On the edge-density of the Brownian co-graphon and common ancestors of pairs in the CRT
本研究给出布朗余图子边密度随机变量矩的简单递推关系，同时研究布朗CRT中k个均匀点对最近公共祖先集合的大小，证明其依概率满足$S_k\sim c\sqrt{k}\log k$（$c=(\sqrt{2π})^{-1}$），并指出相关开放问题。
Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit theory
本文针对单模随机树上交互扩散阈值级联的两个未解决问题，提供分析框架并证明部分定理，涉及前沿传播与连续型极限理论，给出相关猜想与证明策略，未完全解决两个问题。
Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality
本文定义生成树装饰平面图的自然定向，研究有向距离，证明其典型阶为n^{1/3}，Busemann函数标度收敛至3/2-稳定Lévy过程，并推广至√2-LQG普适类，给出有向度量的标度维数。
Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees
本文研究了无标号二叉有根树Colijn-Plazzotta秩与树高的关系，在三种随机模型下推导了秩的渐近均值性质，解决了数学系统发育学中的开放问题。
We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n\rightarrow \infty$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of …
Sampling biased spanning forests through marked vertices
研究完全图K_N的κ偏 spanning forest 中连接多个顶点的子树分布及其渐进行为，并在κ=O(√N)处识别相变现象。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。