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 exploration process and the pruned one.
We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process $\{{\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous process $\{{\cal G}^*(u)\}$ by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process $\{{\cal G}(u)\}$ run until its ascension time has a representation in terms of $\{{\cal G}^*(u)\}$. A similar result was obtained by Aldous and Pitman (1998) in the special cas…
In [Aldous,Pitman,1998] a tree-valued Markov chain is derived by pruning off more and more subtrees along the edges of a Galton-Watson tree. More recently, in [Abraham,Delmas,2012], a continuous analogue of the tree-valued pruning dynamics is constructed along Lévy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of so-called bi-measure trees, which are metric measure spaces with …
We consider a pruning of the inhomogeneous continuum random trees, as well as the cut trees that encode the genealogies of the fragmentations that come with the pruning. We propose a new approach to the reconstruction problem, which has been treated for the Brownian CRT in [Electron. J. Probab. vol. 22, 2017] and for the stable trees in [Ann. IHP B, vol 55, 2019]. Our approach does not rely upon self-similarity and can potentially apply to general Lévy trees as well.
Pruning processes $(\mathcal{F}(θ),θ\geq 0)$ have been studied separately for Galton-Watson trees and for Lévy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem by Abraham and Delmas, also formulated as a conjecture by Löhr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests $\mathcal{F}_n$, $n\geq 1$, in the domain of attraction of a Lévy forest $\mathcal{F}$, suitably scaled pruning processes $(\mathcal{F}_n(θ),θ\geq 0)$ converge in the Skorohod topology on cadlag functions with values in t…
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…
Invariant Galton-Watson (IGW) tree measures is a one-parameter family of critical Galton-Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are the only attractors of critical Galton-Watson measures wit…
Considering a random binary tree with $n$ labelled 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 unable 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 even…
We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li, we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establ…
By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on Lévy trees. Then we could construct a decreasing Lévy-CRT-valued process $\{{\mathcal T}_t\}$ by pruning Lévy trees and an analogous process $\{{\mathcal T}^*_t\}$ by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of $\{{\mathcal T}_t\}$ at the ascension time can be represented by $\{{\mathcal T}^*_t\}$. The results generalize those studied in Abraham and Delmas (2012).
We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree $T$, starting from the leaves, according to a pruning function defined on subtrees within $T$. The generalized pruning encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The main result is invariance of a finite critical binary Galton-Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. The second part of the paper examines the …
It has been claimed in Aldous, Miermont and Pitman [PTRF, 2004] that all Lévy trees are mixings of inhomogeneous continuum random trees. We give a rigorous proof of this claim in the case of a stable branching mechanism, relying on a new procedure for recovering the tree distance from the graphical spanning trees that works simultaneously for stable trees and inhomogeneous continuum random trees.
We prove that the uniform unlabelled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This proves a conjecture by Aldous. Moreover, we establish Benjamini-Schramm convergence of this model of random trees.
We introduce a new stick-breaking construction for inhomogeneous continuum random trees (ICRT). This new construction allows us to prove the necessary and sufficient condition for compactness conjectured by Aldous, Miermont and Pitman arXiv:math/0401115 by comparison with Lévy trees. We also compute the fractal dimensions (Minkowski, Packing, Hausdorff).
We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability of each tree is, in turn, obtained by Möbius inversi…
We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal Hölder exponents of the height functions.
Consider the Aldous--Pitman fragmentation process [Ann Probab, 26(4):1703--1726, 1998] of a Brownian continuum random tree ${\cal T}^{\mathrm{br}}$. The associated cut tree cut$({\cal T}^{\mathrm{br}})$, introduced by Bertoin and Miermont [Ann Appl Probab, 23:1469--1493, 2013], 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 ${\cal T}^{\mathrm{br}}$ to cut$({\cal T}^{\mathrm{br}})$.
We introduce the notion of a restricted exchangeable partition of $\mathbb{N}$. We obtain integral representations, consider associated fragmentations, embeddings into continuum random trees and convergence to such limit trees. In particular, we deduce from the general theory developed here a limit result conjectured previously for Ford's alpha model and its extension, the alpha-gamma model, where restricted exchangeability arises naturally.
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total…
Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the $β(3/2,1/2)$-coalescent. By considering a pruning procedure on stable Galton-Watson tree with $n$ labeled leaves, we give a representation of the discrete $β(1+α,1-α)$-coalescent, with $α\in [1/2,1)$ starting from the trivial partition of the $n$ first integers. The construction can also be made directly on the stable continuum L{é}vy tree, with parameter $1/α$, simultaneously for all $n$. This representation allows to use …
We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to es…
By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $Θ$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.
Consider a Markov chain on the space of rooted real binary trees that randomly removes leaves and reinserts them on a random edge and suitably rescales the lengths of edges. This chain was introduced by David Aldous who conjectured a diffusion limit of this chain, as the size of the tree grows, on the space of continuum trees. We prove the existence of a process on continuum trees, which via a random time change, displays properties one would expect from the conjectured Aldous diffusion. The existence of our process is proved by considering an explicit scaled limit of a Poissonized version of …
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the structure of an ICRT, and also of its width process, determining the size of layers in order of height. These processes turn out to be transformations of bridges with exchangeable increments, which have already appeared in other ICRT related topics such as stochastic additive coalescence. The results rely on two different constructions of birthday trees from processes with exchangeable increments, on…
We introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The operations are based on spinal decompositions and a regenerativity preserving merging procedure of $(α, θ)$-strings of beads, that is, random intervals $[0, L_{α, θ}]$ equipped with a random discrete measure $dL^{-1}$ arising in the limit of ordered $(α, θ)$-Chinese restaurant processes as introduced recently by Pitman and Winkel. Indeed, we iterate the branch mergin…
Given a general critical or sub-critical branching mechanism and its associated Lévy continuum random tree, we consider a pruning procedure on this tree using a Poisson snake. It defines a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. This work generalizes the work made for a Brownian tree [Abraham, Serlet] and for a tree without Brownian part [Abraham, Delmas].
Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous's beta-splitting models and Ford's alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.
Self-similarity of random trees is related to the operation of pruning. Pruning $R$ cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex $v$ and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at $v$. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers $N_k[K]$ and $N_{ij}[K]$ are defined as the expected number of branches of order $k$, and the expected number of order-$i$ branches that merged order-$j$…
The Aldous diffusion is a conjectured Markov process on the space of real trees that is the continuum analogue of discrete Markov chains on binary trees. We construct this conjectured process via a consistent system of stationary evolutions of binary trees with $k$ labeled leaves and edges decorated with diffusions on a space of interval partitions constructed in previous work by the same authors. This pathwise construction allows us to study and compute path properties of the Aldous diffusion including evolutions of projected masses and distances between branch points. A key part of proving t…
We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary converges in distribution to a continuum random tree with respect to Gromov Hausdorff topology. This verifies a prediction made by Aldous for a simpler model (where paths are independent till they coalesce) but for a different conditional set up. The limiting continuum tree is slightly different from what was surmised earlier. Our proof uses the fact that there ex…
In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost-surely logarithmic global fluctuations and log-logarithmic local fluctuations in the volume of balls of radius $r$ about the leading order polynomial term as $r\to0$. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as $t\to0$ almost-surely. Finally, we prove that this quenched (almost-sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) vo…
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…
To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the $k$-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the $k$-cut model to isolate the root of a Galton--Watson tree with a finite-variance offspring law and conditioned to have $n$ nodes, when divided by $n^{1-1/2k}$, converges in distribution to some random variable defined on the Brownian CRT. We provide here a direct construction of the limit random variable, relying upon th…
We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson--Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees a…
We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay. We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of…
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2003) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of ``attracting points'' to emerge.
We study a genealogical model for continuous-state branching processes with immigration with a (sub)critical branching mechanism. This model allows the immigrants to be on the same line of descent. The corresponding family tree is an ordered rooted continuum random tree with a single infinite end defined thanks to two continuous processes denoted by $(\overleftarrow{H}_t ;t\geq 0)$ and $(\overrightarrow{H}_t ;t\geq 0)$ that code the parts at resp. the left and the right hand of the infinite line of descent of the tree. These processes are called the left and the right height processes. We defi…
We perform a pruning procedure on a Lévy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the Lévy tree.
Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+ε$ (resp. $1/2 -ε$), $ε\in (0, 1/2)$, and choosing direction $i\in \{1, ..., b\}$ when going up with probability $a_i$. Here $å=(a_1, ..., a_b)$ stands for some non-degenerated fixed set of weights. We consider the range $\{W^{\ee}_n ; n\geq 0 \}$ that is a subtree of $\U_b $. It corresponds to a unique random rooted ordered tree that we denote by $τ_ε$. We rescale the edges of $τ_ε$ by a factor $\ee $ …
The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding. Ubiquitous in hierarchical branching systems, the Horton laws have been rediscovered in many disciplines ranging from geomorphology to genetics to computer science. Attempts to build a mathematical foundation behind the Horton laws during the 1990s revealed their close connection to the operation of pruning -- erasing a tree from the leaves down to the root. This survey synthesizes recent results on invariances and self-similarities of tree measures under vari…
We give a description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton-Watson tree and a one-parameter family of critical Galton-Watson trees with offspring distribution $\{q_k\}$ that has a power tail $q_k\sim Ck^{-(1+1/q_0)}$, where $q_0\in(1/2,1)$. Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the ta…
We consider a family of discrete coagulation-fragmentation equations closely related to the one-dimensional forest-fire model of statistical mechanics: each pair of particles with masses $i,j \in \nn$ merge together at rate 2 to produce a single particle with mass $i+j$, and each particle with mass $i$ breaks into $i$ particles with mass 1 at rate $(i-1)/n$. The (large) parameter $n$ controls the rate of ignition and there is also an acceleration factor (depending on the total number of particles) in front of the coagulation term. We prove that for each $n\in \nn$, such a model has a unique eq…
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.
We introduce a new phylogenetic reconstruction algorithm which, unlike most previous rigorous inference techniques, does not rely on assumptions regarding the branch lengths or the depth of the tree. The algorithm returns a forest which is guaranteed to contain all edges that are: 1) sufficiently long and 2) sufficiently close to the leaves. How much of the true tree is recovered depends on the sequence length provided. The algorithm is distance-based and runs in polynomial time.
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldo…
We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the ICRTs (scaling limits of $\mathbf p$-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation. The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree.
We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.
The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be "equivalent" to the Brownian map but the exact nature of the correspondence (i.e.\ embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree…
