Brownian Continuum Random Tree Research Articles

Coupling Bertoin's and Aldous–Pitman's representations of the additive coalescent

AbstractWe construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous and Pitman, involves the components obtained by logging the Brownian continuum random tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running infimum of a linear‐drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so‐called cut‐tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT.

Cutoff on Trees is Rare

We study the simple random walk on trees and give estimates on the mixing and relaxation times. Relying on a seminal result by Basu, Hermon and Peres characterizing cutoff on trees, we give geometric criteria that are easy to verify and allow to determine whether the cutoff phenomenon occurs. We provide a general characterization of families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton–Watson trees of a fixed height, and sequences of random trees converging to the Brownian continuum random tree.

Stable graphs: distributions and line-breaking construction

For $\alpha \in (1,2]$, the $\alpha$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $\alpha$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $\mathbb R$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $\mathbb R$-tree, which is a biased version of the $\alpha$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the $\alpha$-stable graph as a collection of $\alpha$-stable trees glued onto its kernel and (3) present a line-breaking construction, in the same spirit as Aldous' line-breaking construction of the Brownian continuum random tree.

On breadth‐first constructions of scaling limits of random graphs and random unicellular maps

We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map of a given genus that start with a suitably tilted Brownian continuum random tree and make “horizontal” point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth-first construction of a finite connected graph. In particular, this yields a breadth-first construction of the scaling limit of the critical Erdős–Rényi random graph which answers a question posed by Addario-Berry, Broutin, and Goldschmidt. As a consequence of this breadth-first construction, we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions.

A geometric representation of fragmentation processes on stable trees

We provide a new geometric representation of a family of fragmentation processes by nested laminations which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation, obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous–Pitman fragmentation of the Brownian continuum random tree and minimal factorizations of the n-cycle, that is, factorizations of the permutation (12⋯n) into a product of (n−1) transpositions, proving this way a conjecture of Féray and Kortchemski. We discuss various properties of these new lamination-valued processes, and we notably show that they can be coded by explicit Lévy processes.

The distance profile of rooted and unrooted simply generated trees

AbstractIt is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, 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 any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

Brownian motion on stable looptrees

In this article, we introduce Brownian motion on α-stable looptrees using resistance techniques, where α∈(1,2). We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of rα and t−αα+1 respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

On quasisymmetric embeddings of the Brownian map and continuum trees

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 into {{,mathrm{mathbb {R}},}}^d, {{,mathrm{mathbb {S}},}}^d, {{,mathrm{mathbb {T}},}}^d, or more generally any doubling metric space, cannot be quasisymmetric. We achieve this with the aid of dimension theory by identifying a metric structure that is invariant under quasisymmetric mappings (such as isometries) and which implies infinite Assouad dimension. We show, using elementary methods, that this structure is almost surely present in the Brownian continuum random tree and the Brownian map. We further show that snowflaking the metric is not sufficient to find an embedding and discuss continuum trees as a tool to studying “fractal functions”.

K-cut model for the Brownian continuum random tree

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 n1−1∕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 the Aldous–Pitman fragmentation process and a deterministic time change.

Scaling limit of random forests with prescribed degree sequences

In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov–Hausdorff–Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov–Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree (BCRT), pioneered by the work of Aldous (Ann. Probab. 19 (1991) 1–28; In Stochastic Analysis (Durham, 1990) (1991) 23–70 Cambridge Univ. Press; Ann. Probab. 21 (1993) 248–289). In fact, we identify the limiting random object as a sequence of random real trees encoded by excursions of some first passage bridges reflected at minimum. We establish such convergence by studying the associated Lukasiewicz walk of the degree sequences. In particular, our work is closely related to and uses the results from the recent work of Broutin and Marckert (Random Structures Algorithms 44 (2014) 290–316) on scaling limit of random trees with prescribed degree sequences, and the work of Addario-Berry (Random Structures Algorithms 41 (2012) 253–261) on tail bounds of the height of a random tree with prescribed degree sequence.

On the Brownian separable permuton

AbstractThe Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

Inverting the cut-tree transform

Nous considérons des fragmentations d’un $\mathbb{R}$-arbre $T$ dirigées par des coupures qui arrivent selon un processus de Poisson sur $T\times[0,\infty)$, où la première composante désigne le point auquel se produit la coupure et le deuxième, l’instant auquel elle a lieu. La généalogie d’une telle fragmentation est codée par l’arbre des coupes, qui a été introduit par Bertoin et Miermont (Ann. Appl. Probab. 23 (4) (2013) 1469–1493) pour une fragmentation de l’arbre brownien. L’arbre des coupes a ensuite été généralisé par Dieuleveut (Ann. Appl. Probab. 25 (4) (2015) 2215–2262) à une fragmentation des arbres $\alpha$-stables, pour $\alpha\in(1,2)$, et par Broutin et Wang (Bernoulli 23 (4A) (2017) 2380–2433) aux arbres continus inhomogènes d’Aldous et Pitman (Probab. Theory Related Fields 118 (4) (2000) 455–482). Dans les deux premiers cas, l’évolution de la suite des masses des sous-arbres apparaissant dans le processus de fragmentation constitue une famille importante de processus de fragmentation auto-similaires de Bertoin (Ann. Inst. Henri Poincaré Probab. Stat. 38 (3) (2002) 319–340); dans le premier cas, une fois inversée dans le temps, la fragmentation devient un coalescent additif. Remarquablement, dans tous ces cas, la loi de l’arbre des coupes est la même que celle du $\mathbb{R}$-arbre initial. Dans cet article, nous développons un cadre général pour l’étude de l’arbre des coupes d’un $\mathbb{R}$-arbre. Par la suite, nous nous concentrons particulièrement sur le problème de la reconstruction : comment retrouver le $\mathbb{R}$-arbre initial à partir de son arbre des coupes. Ce problème a été étudié pour l’arbre brownien par Broutin et Wang (Electron. J. Probab. 22 (2017) 80), qui démontrent qu’il est possible de reconstruire l’arbre initial en loi. Nous décrivons un enrichissement de la construction de l’arbre des coupes, qui dote l’arbre des coupes d’une structure supplémentaire que nous appelons une collection cohérente de routages. Nous démontrons que ce procédé est bien défini sous des conditions minimales sur le $\mathbb{R}$-arbre. Ensuite, nous démontrons que pour l’arbre brownien ainsi que pour l’arbre $\alpha$-stable avec $\alpha\in(1,2)$, l’arbre initial muni de son processus de Poisson de coupures peut être reconstruit presque sûrement à partir de l’arbre des coupes enrichi. Pour ces derniers résultats, nous utilisons de façon essentielle l’auto-similarité et l’invariance par réenracinement de ces $\mathbb{R}$-arbres aléatoires.

Heavy subtrees of Galton-Watson trees with an application to Apollonian networks

We study heavy subtrees of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size being $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is among the $k$ heaviest nodes among its siblings. Unmarked nodes and their subtrees are removed, leaving only a tree of marked nodes, which we call the $k$-heavy tree. We study various properties of these trees, including their size and the maximal distance from any original node to the $k$-heavy tree. In particular, under some moment condition, the $2$-heavy tree is with high probability larger than $cn$ for some constant $c > 0$, and the maximal distance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As a consequence, for uniformly random Apollonian networks of size $n$, the expected size of the longest simple path is $\Omega (n)$. We also show that the length of the heavy path (that is, $k=1$) converges (after rescaling) to the corresponding object in Aldous’ Brownian continuum random tree.

The continuum random tree is the scaling limit of unlabeled unrooted trees

We show that the uniform unlabeled 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 confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

We study a fragmentation of the p-trees of Camarri and Pitman. We give exact correspondences between the p-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the inhomogeneous continuum random trees (scaling limits of p-trees) and give distributional correspondences between the initial tree and the tree encoding the fragmentation. The theorems for the inhomogeneous continuum random tree extend previous results by Bertoin and Miermont about the cut tree of the Brownian continuum random tree.

Central limit theorems for the spectra of classes of random fractals

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known.

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

Scaling limits of random graphs from subcritical classes

We study the uniform random graph C-n with n vertices drawn from a subcritical class of connected graphs. Our main result is that the resealed graph C-n / root n converges to the Brownian continuum random tree T-e multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide sub-Gaussian tail bounds for the diameter D (C-n) and height H(C-n(center dot)) of the rooted random graph C-n(center dot) We give analytic expressions for the scaling factor in several cases, including for example the class of outerplanar graphs. Our methods also enable us to study first passage percolation on C-n, where we also show the convergence to T-e under an appropriate rescaling.

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.