Continuum Random Tree Research Articles

Mutational pattern of a sample from a critical branching population.

We study a universal object for the genealogy of a sample in populations with mutations: the critical birth-death process with Poissonian mutations, conditioned on its population size at a fixed time horizon. We show how this process arises as the law of the genealogy of a sample in a large class of nearly critical branching populations with rare mutations at birth, namely populations converging, in a large population asymptotic, towards the continuum random tree. We extend this model to populations with random foundation times, with (potentially improper) prior distributions [Formula: see text], [Formula: see text], including the so-called uniform ([Formula: see text]) and log-uniform ([Formula: see text]) priors. We first investigate the mutational patterns arising from these models, by studying the site frequency spectrum of a sample with fixed size, i.e. the number of mutations carried by k individuals in the sample. Explicit formulae for the expected frequency spectrum of a sample are provided, in the cases of a fixed foundation time, and of a uniform and log-uniform prior on the foundation time. Second, we establish the convergence in distribution, for large sample sizes, of the (suitably renormalized) tree spanned by the sample with prior [Formula: see text] on the time of origin. We finally prove that the limiting genealogies with different priors can all be embedded in the same realization of a given Poisson point measure.

Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes

Some, but not all processes of the form $M_{t}=\exp(-\xi_{t})$ for a pure-jump subordinator $\xi$ with Laplace exponent $\Phi$ arise as residual mass processes of particle 1 (tagged particle) in Bertoin’s partition-valued exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of $M=(M_{t},t\ge0)$ in a fragmentation process, and we show that for each $\Phi$, there is a unique (in distribution) binary fragmentation process in which $M$ has a Markovian embedding. The identification of the Laplace exponent $\Phi^{*}$ of its tagged particle process $M^{*}$ gives rise to a symmetrisation operation $\Phi\mapsto\Phi^{*}$, which we investigate in a general study of pairs $(M,M^{*})$ that coincide up to a random time and then evolve independently. We call $M$ a fragmenter and $(M,M^{*})$ a bifurcator. For $\alpha>0$, we equip the interval $R_{1}=[0,\int_{0}^{\infty}M_{t}^{\alpha}\,dt]$ with a purely atomic probability measure $\mu_{1}$, which captures the jump sizes of $M$ suitably placed on $R_{1}$. We study binary tree growth processes that in the $n$th step sample an atom (“bead”) from $\mu_{n}$ and build $(R_{n+1},\mu_{n+1})$ by replacing the atom by a rescaled independent copy of $(R_{1},\mu_{1})$ that we tie to the position of the atom. We show that any such bead splitting process $((R_{n},\mu_{n}),n\ge1)$ converges almost surely to an $\alpha$-self-similar continuum random tree of Haas and Miermont, in the Gromov–Hausdorff–Prohorov sense. This generalises Aldous’s line-breaking construction of the Brownian continuum random tree.

Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set

Nous obtenons des limites d’échelle pour des arbres branchants markoviens dont la taille est égale au nombre de noeuds dont le degré sortant appartient à un ensemble fixé. Ceci étend des résultats récents de Haas et Miermont dans (Ann. Probab. 40 (2012) 2589–2666), qui ont considéré le cas où la taille d’un arbre est le nombre de ses feuilles ou le nombre des sommets. Nous utilisons nos résultats pour prouver que la limite d’échelle d’arbres de Galton–Watson conditionnés par le nombre de noeuds dont le degré sortant appartient à un ensemble donné est l’arbre brownien continu. La clé pour appliquer notre résultat pour les arbres branchants markoviens à des arbres de Galton–Watson conditionnés est une généralisation de la formule classique de Otter–Dwass. Ceci est obtenu en montrant que le nombre de sommets d’un arbre de Galton–Watson dont le degré sortant appartient à un ensemble donné est distribué comme le nombre de sommets dans un arbre de Galton–Watson avec une loi de reproduction appropriée.

Decomposition of Lévy trees along their diameter

We study the diameter of Levy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Levy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Levy trees and we prove that it is realized by a unique pair of points. We prove that the law of Levy trees conditioned to have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their respective roots two independent size-biased Levy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Levy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Levy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case.

Schröder’s problems and scaling limits of random trees

In his now classic paper of 1870, Schröder posed four combinatorial problems about the number of certain types of bracketings of words and sets. Here we address what these bracketings look like on average. For each of the four problems we prove that a uniform pick from the appropriate set of bracketings, when considered as a tree, has the Brownian continuum random tree as its scaling limit as the size of the word or set goes to infinity.

A line-breaking construction of the stable trees

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

Scaling Limits of Random Graphs from Subcritical Classes: Extended abstract

We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling. On s’int´eresse au comportement asymptotique du graphe aleatoire $\mathsf{C}_n$ sur $n$ sommets pris uniformément d’une classe sous-critique des graphes sur n sommets. Dans cette contribution nous montrons que le graphe normalisée$\mathsf{C}_n / \sqrt{n}$ converges vers un arbre aléatoire brownien continue Te multiplie par une constante qui dépends de la classede graphes considérée. Nous calculons l’expression analytique pour cette constante dans plusieurs cas parmi la classefameuse des graphes planaire extérieure. En plus, on montre que le diamètre $\text{D}(\mathsf{C}_n)$ et la hauteur $\text{H}(\mathsf{C}_n^\bullet)$ de l’équivalent racine de $\mathsf{C}_n$ sont bornes par des bornes sous gaussiens. Notre méthode nous permettons aussi de l’étudier la percolation du premier passage sur $\mathsf{C}_n$. Nous montrons que $\mathcal{T}_{\mathsf{e}}$ sujet a une changement d’échelle appropriée

The Brownian continuum random tree as the unique solution to a fixed point equation

© 2015, University of Washington. All rights reserved. In this note, we provide a new characterization of Aldous’ Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes without replacement [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact tree-like random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$. Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in [14] and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. Keywords: Multiplicative coalescent, p-trees, inhomogeneous continuum random trees, critical random graphs, Gromov-Hausdorff distance, Gromov-weak topology

Reversing the cut tree of the Brownian continuum random tree

Consider the Aldous–Pitman fragmentation process of a Brownian continuum random tree T^{br}. The associated cut tree cut(T^{br}), introduced by Bertoin and Miermont, is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking T br to cut(T^{br}).

Exceptionally small balls in stable trees

The $\gamma$-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $\gamma$-stable domain, $\gamma \in (1, 2]$. They form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in1998) and the Brownian case $\gamma= 2$ corresponds to Aldous Continuum Random Tree (CRT). In this paper, we study fine properties of the mass measure, that is the natural measure on $\gamma$-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 $r^{\frac{\gamma}{\gamma-1}} (\log1/r)^{-\frac{1}{\gamma-1}}$. We think that no similar result holds true for the maximum of the mass measure of balls with radius $r$, except in the Brownian case: when $\gamma = 2$, we prove that this quantity is of order $r^2 \log 1/r$. In addition, we compute the exact constant for the lower local density of the mass measure (and the upper one for the CRT), which continues previous results.

Dissecting the circle, at random

Random laminations of the disk are the continuous limits of random non-crossing congurations of regular polygons. We provide an expository account on this subject. Initiated by the work of Aldous on the Brownian triangulation, this eld now possesses many characters such as the random recursive triangulation, the stable laminations and the Markovian hyperbolic triangulation of the disk. We will review the properties and constructions of these objects as well as the close relationships they enjoy with the theory of continuous random trees. Some open questions are scattered along the text.

A Construction of a β-Coalescent via the Pruning of Binary Trees

Considering a random binary tree withnlabelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

A Construction of a β-Coalescent via the Pruning of Binary Trees

Considering a random binary tree withnlabelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees

We introduce the notion of a restricted exchangeable partition of N and study natural classes of such partitions. We obtain integral representations, study associated coalescents and fragmentations, embeddings into continuum random trees and convergence to such limit trees. As an application, we deduce from the general theory developed here a particular limit result conjectured previously for Ford’s alpha model and its non-binary extension, the alpha-gamma model, where restricted exchangeability arises naturally. AMS 2000 subject classifications: 60G09, 60J80.

The cut-tree of large Galton–Watson trees and the Brownian CRT

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 theorem due to Janson [Random Structures Algorithms 29 (2006) 139-179] for the number of random cuts needed to isolate the root in Galton-Watson trees conditioned by their sizes, and also generalizes a recent result [Ann. Inst. Henri Poincar\'{e} Probab. Stat. (2012) 48 909-921] obtained in the special case of Cayley trees.

The scaling limit of the minimum spanning tree of the complete graph

Consider the minimum spanning tree (MST) of the complete graph with nn vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n1/3n1/3 and with the uniform measure on its vertices. We show that the resulting space converges in distribution as n→∞n→∞ to a random compact measured metric space in the Gromov–Hausdorff–Prokhorov topology. We additionally show that the limit is a random binary RR-tree and has Minkowski dimension 33 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erdős–Renyi random graph. We exploit the explicit description of the scaling limit of the Erdős–Renyi random graph in the so-called critical window, established in [Probab. Theory Related Fields 152 (2012) 367–406], and provide a similar description of the scaling limit for a “critical minimum spanning forest” contained within the MST. In order to accomplish this, we introduce the notion of RR-graphs, which generalise RR-trees, and are of independent interest.

Asymptotics of trees with a prescribed degree sequence and applications

AbstractLettbe a rooted tree andnbi(t) the number of nodes inthavingichildren. The degree sequenceoftsatisfies, wheredenotes the number of nodes int. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence; we writefor the corresponding distribution. Letbe a list of degree sequences indexed by κ corresponding to trees with size. We show that under some simple and natural hypotheses onthe trees sampled underconverge to the Brownian continuum random tree after normalisation by. Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014

Scaling Limit for the Random Walk on the Largest Connected Component of the Critical Random Graph

A scaling limit for the simple random walk on the largest connected component of the Erd\H{o}s-R\'{e}nyi random graph G(n,p) in the critical window, p=n^{-1}+\lambda n^{-4/3} , is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.

A continuum-tree-valued Markov process

We present a construction of a L\'evy 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 $\psi$ be a critical branching mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing L\'evy-CRT-valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and super-critical if $\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.