Brigitte Chauvin, Thierry Klein, Jean-Francois Marckert, Alain Rouault
We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.
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We present new links between some remarkable martingales found in the study of the Binary Search Tree, or of the Bisection Problem, looking at them on the probability space of a continuous time binary branching process.
We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [9]. Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [17]. Surprisingly, this phase transition phenomenon disappears in the sense …
Broadcasting on Random Trees: Martingale Methods and Error Bounds
针对随机树上广播过程中根节点颜色的重构问题，提出一种鞅方法分析颜色不平衡性，统一处理不同树模型，给出多数规则估计器误差概率的线性上界。
We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, agrees up to generation $K$ with a regular $μ$-ary tree, where $μ$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $\log(1/\eps)$. More precisely, we show that if $μ\ge 2$ then with high probability as $\eps \downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $μ$-ary tree, providing an ex…
For a martingale $(X_n)$ converging almost surely to a random variable $X$, the sequence $(X_n - X)$ is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of R{é}gnier's martingale for the path length in random binary search trees. Gr{ü}bel and Kabluchko [to appear in Annals of Applied Probability, (2016), arXiv 1410.0469] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the…
Penalization of Galton Watson trees with marked vertices
针对节点按出度概率独立标记的 Galton-Watson 树，本文通过惩罚方法构造含层标记数的新鞅，并经 Girsanov 变换定义新测度，刻画随机树的新分布。
About the asymptotic behaviour of the martingale associated with the Vertex Reinforced Jump Process on trees and Z d
本文研究树上和 Z^d 上 VRJP 相关鞅的渐近行为，刻画其 L^p 有界性、一致可积性、常返与临界衰减率，并证明临界点对应正常返 Markov 链混合。
In this paper we prove a functional limit theorem for the weighted profile of a $b$-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.
We establish a central limit theorem for (a sequence of) multivariate martingales which dimension potentially grows with the length $n$ of the martingale. A consequence of the results are Gaussian couplings and a multiplier bootstrap for the maximum of a multivariate martingale whose dimensionality $d$ can be as large as $e^{n^c}$ for some $c>0$. We also develop new anti-concentration bounds for the maximum component of a high-dimensional Gaussian vector, which we believe is of independent interest. The results are applicable to a variety of settings. We fully develop its use to the estimation…
Tree Coordinates and Range Martingales for Positive Operator-Valued Measures
研究正算子值测度在树上的内在局部坐标，通过局部分裂恢复测度并构建扩张，其换位子成为值域鞅演算，给出极值和控制的局部描述及有界测度变换公式，自伴值域鞅有二次变差公式。
The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with t…
Total Path Length in Power-Weight Recursive Trees: Martingale Limits and Global Fluctuations
本研究在幂权递归树中建立了总路径长度的鞅极限与方差渐近，覆盖所有实数幂指数，并揭示与随机二叉搜索树在全局波动上的差异。
Null models of binary phylogenetic trees are useful for testing hypotheses on real world phylogenies. In this paper we consider phylogenies as binary trees without edge lengths together with a sampling measure and encode them as algebraic measure trees. This allows to describe the degree of similarity between actual and simulated phylogenies by focusing on the sample shape of subtrees and their subtree masses. We describe the annealed law of the statistics of subtree masses of null models, namely the branching tree, the coalescent tree, and the comb tree in more detail. Finally, we use methods…
We consider directed polymers in random environment on the lattice Z d at small inverse temperature and dimension d $\ge$ 3. Then, the normalized partition function W n is a regular martingale with limit W. We prove that n (d--2)/4 (W n -- W)/W n converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale W n are different from those for polymers on trees.
Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploratio…
A limit law for the most favorite point of simple random walk on a regular tree
本文研究了有限深度正则树上连续时间随机游走在叶子顶点中的最偏爱点，证明了在深度趋于无穷大时，经过缩放和居中的最大停留时间收敛至随机平移的 Gumbel 律，并通过导数鞅类对象刻画了该随机平移。
We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees.
We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who showed that $X(T_n)/\sqrt{2n}$ converges a.s. towards a Rayleigh-distributed random variable $Θ$, which gives a continuous analog to an earlier result by Janson on conditioned, finite-variance Galton-Watson trees. We prove a convergence in distribution of $n^{-1/4}(X(T_n)-\sqrt{2n}Θ)$ towards…
Let $W_{\infty}(β)$ be the limit of the Biggins martingale $W_n(β)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of bina…
We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all…
We revisit the random $m$-ary search tree and study a finer profile of its node outdegrees with the purpose of exploring possibilities of data structure compression. The analysis is done via Pólya urns. The analysis shows that the number of nodes of each individual node outdegree has a phase transition: Up to $m=26$, the number of nodes of outdegree $k$, for $k=0,1, \ldots, m$, is asymptotically normal; that behavior changes at $m = 27$. Based on the analysis, we propose a compact $m$-ary tree that offers significant space saving.
Depth profile of depth-weighted trees with bounded weights
研究权重有界的深度加权随机递归树，通过定义鞅结合Edgeworth展开，在权重满足大数定律时确定树深度分布的缩放极限，推广相关结果并否定一个未决问题。
Drmota and Gittenberger (1997) proved a conjecture due to Aldous (1991) on the height profile of a Galton-Watson tree with an offspring distribution of finite variance, conditioned on a total size of $n$ individuals. The conjecture states that in distribution its shape, more precisely its scaled height profile coincides asymptotically with the local time process of a Brownian excursion of duration 1. We give a proof of the result, which extends to the case of an infinite variance offspring distribution. This requires a different strategy, since in the infinite variance case there is no longer …
Markov properties for the vertical edge profile in random labelled trees
研究随机标记树中垂直边轮廓的马尔可夫性质，揭示其与超布朗运动的关联。
We give a detailed asymptotic analysis of the profiles of random symmetric digital search trees, which are in close connection with the performance of the search complexity of random queries in such trees. While the expected profiles have been analyzed for several decades, the analysis of the variance turns out to be very difficult and challenging, and requires the combination of several different analytic techniques, including Mellin and Laplace transforms, analytic de-Poissonization, and Laplace convolutions. Our results imply concentration of the profiles in the range where the mean tends t…
We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloorα\log n\rfloor$ in a certain range of $α$. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the…
We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned about counts of the number of vertices of a particular containment level. For the local containment profile, we obtain the exact mean, variance and probability distribution in terms of standard combinatorial quantities like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an as…
We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by $2^{n+1} n$ and then centered by $(\log 2) n - \log n$, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated di…
The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W\egalLoi\sum_{k=1}^mV_k^λW_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random v…
A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{\bf s}$ on rooted plane trees whose degree sequence equals ${\bf s}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (19…
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment.
Maximum Spanning Trees of Random Geometric Graphs With Independent Edge Weights
本文针对具有独立边权重的随机几何图，通过分割与迭代路径构造法得到最大生成树权重增长的偏差界，利用鞅差方法确定最大权重L²收敛的充分条件。
Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds
本文研究了三进制鞅等周轮廓函数，证明了其为Takagi型Bellman函数，并给出了n-adic情况下鞅的下界及端点估计。
Conductance of a subdiffusive random weighted tree
本文研究次扩散区域内随机加权 Galton–Watson 树的根到第 n 层电导衰减，利用解析方法揭示其期望可小于 1/n，并证明归一化电导向加法鞅极限收敛。
Applying affine urn models to the global profile of hyperrecursive trees
本文利用仿射瓮模型研究超递归树的全局包含轮廓，推导前两个混合矩的渐近演化并建立多元中心极限定理，模拟验证了理论结果。
Diffusions and random walks with prescribed sub-Gaussian heat kernel estimates
本文提出了一种新的扩散构造，用于研究具有预设亚高斯热核估计的随机游走和扩散过程。
We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees.
Gaussian free field on the tree subject to a hard wall II: Asymptotics
本文在叶子均为正的硬壁条件下，利用前作确定的排斥剖面，给出二叉树上离散高斯自由场局部与全局统计量的精确渐近描述，并证明条件场与无条件场的分布律渐近相互奇异。
Extremal Spanning Trees of Random Marked Graphs with Independent Edge Weights
本文研究带独立边权和顶点标记的伯努利随机图的极值生成树代价，发现基于尾部指数存在相变，并用鞅差方法证明缩放中心化后代价的L^2收敛。
It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of…
A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier ti…
The height of the infection tree
本文研究了随机SIR模型中感染树的高度，证明在感染未迅速灭绝的情况下，当初始健康个体数趋于无穷时，树高渐近于$κ(λ) \log n$，并发现$κ(λ)$在$λ_{c}\simeq 1.8038$处发生二阶相变。
A particular continuous-time multitype branching process is considered, it is the continuous-time embedding of a discrete-time process which is very popular in theoretical computer science: the m-ary search tree (m is an integer). There is a well-known phase transition: when m \leq 26, the asymptotic behavior of the process is Gaussian, but for m \geq 27 it is no more Gaussian and a limit W of a complex-valued martingale arises. Thanks to the branching property it appears as a solution of a smoothing equation of the type Z = e^{-λT}(Z(1) + ... + Z(m)), where λ \in C, the Z(k) are independent c…
We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $μ$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\T^nμ)_{n\ge 0}$ (this completes in a non trivial way our previous result in th…
We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to th…
Tree Capacity and Splitting Isometries for Subinvariant Kernels
本文研究子不变核的树容量与分裂等距映射，通过电网络转换对角增长为电阻和容量，提出显式条件和定量界限，并获得不变完成的最小性性质。
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.
The generalized Zagreb index for non-plane and plane recursive trees
本文研究非平面与平面递归树的广义Zagreb指数，采用矩转移方法证明其矩的一阶渐近性及极限定律，发现非平面树极限为正态分布，平面树三次和四次幂为非正态分布。
Every cutoff profile is possible
本文提出果实合生树马尔可夫链这一参数可调的构造来源，并证明其可实现任意截止剖面、截止时间及窗口大小。
We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886--918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (3--4): 367--40…
Critical scaling profile for trees and connected subgraphs on the complete graph
通过分析完全图上树和连通子图的生成函数，在临界窗口内识别出两者共有的单一标度轮廓，并推测该轮廓适用于高维离散环面（d≥8）上的格点树和格点动物。
Scaling limit of trees with vertices of fixed degrees and heights
本文研究了度数和高度固定的树的标度极限，通过凝聚过程分析路径收敛性，并应用于变化环境下的 Bienaymé-Galton-Watson 树。
We show that the profile of the tree constructed by the Depth First Search Algorithm in the giant component of an Erdős-Rényi graph with $N$ vertices and connection probability $c/N$ converges to an explicit deterministic shape. This makes it possible to exhibit a long non-intersecting path of length $\left( ρ_c - \frac{\mathrm{Li}_2(ρ_c)}{c} \right) \times N$, where $ρ_c$ is the density of the giant component.
We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when $n$ goes to infinity, a random $2\ka$-angulation with $n$ faces has a separating vertex whose removal disconnects the map into two components each with s…
Depths in random recursive metric spaces
推广随机递归树与优先连接树，提出随机递归度量空间，利用鞅理论证明插入深度的大数定律与中心极限定理，并应用于随机树、钩连网络及连续空间的推广。
A random recursive tree model with doubling events
提出一种新的随机树模型，在随机递归树基础上引入全局加倍事件，研究其大小、度分布、高度轮廓的渐近性质，并给出高度下界。
From Cannings model to Brownian motion conditioned on local time profile
本文研究了Cannings模型谱系树的缩放极限，证明了其收敛于局部时间剖面条件下的布朗运动，并利用从无穷下降性质进行证明。
A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzman…
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 …
Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures $μ_{n,β}, n = 1,2,\dots$, parameterized by a positive disorder parameter $β$, and defined on the Borel $σ$-field ${\mathcal B}$ of $\partial T = \{0,1,\dots b-1\}^\infty$ for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures $prob_{n,β}:= Z_{n,β}^{-1}μ_{n,β}, n = 1,2\dots,$ normalized to a probability by the partition function $Z_{n,β}$. In this note, a recent result of Madaule (2011) is used to explicitly construct a family o…
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