Continuum Random Tree Research Articles

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.

Brownian disks and the Brownian snake

Nous donnons une nouvelle construction des disques browniens, qui ont ete definis par Bettinelli et Miermont comme limites d’echelle de quadrangulations avec frontiere quand la taille de la frontiere tend vers l’infini. Notre methode est semblable a la construction de la carte brownienne, mais elle utilise la mesure d’excursion positive du serpent brownien introduite recemment. Cette mesure d’excursion implique un arbre aleatoire continu dont les sommets recoivent des labels positifs, qui correspondent aux distances depuis la frontiere dans notre approche du disque brownien. Nous donnons plusisurs applications de cette construction. En particulier, nous montrons que la mesure uniforme sur la frontiere peut etre obtenue comme limite de la mesure de volume (convenablement normalisee) sur un petit voisinage tubulaire de la frontiere. Nous montrons aussi que les composantes connexes du complementaire du filet brownien sont des disques browniens, comme cela est suggere dans le travail recent de Miller et Sheffield. Finalement, nous montrons que les composantes connexes du complementaire d’une boule centree au point distingue de la carte brownienne sont, conditionnellement a leurs volumes et leurs perimetres, des disques browniens independants.

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.

Harmonic functions on mated-CRT maps

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma $-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos (\pi \gamma ^{2}/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties of random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps — including mated-CRT maps and the UIPT — the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_{n}(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.

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.

Graph limits of random graphs from a subset of connected k-trees.

For any set Ω of non‐negative integers such that {0,1}⊊Ω, we consider a random Ω‐k‐tree G n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that Gn,k, scaled by (kHkσΩ)/(2n) where H k is the kth harmonic number and σ Ω > 0, converges to the continuum random tree Te. Furthermore, we prove local convergence of the random Ω‐k‐tree Gn,k∘ to an infinite but locally finite random Ω‐k‐tree G∞,k.

Recursive construction of continuum random trees

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 Haas, Miermont and Stephenson, and we give for the first time constructions of random compact $\mathbb{R}$-trees that describe the genealogies of Bertoin’s self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.

An invariance principle for branching diffusions in bounded domains

We study branching diffusions in a bounded domain $D$ of $\mathbb{R}^d$ in which particles are killed upon hitting the boundary $\partial D$. It is known that any such process undergoes a phase transition when the branching rate $\beta$ exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous' Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.

Branching processes seen from their extinction time via path decompositions of reflected Lévy processes

We consider a spectrally positive Levy process $X$ that does not drift to $+\infty $, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf _{[0,t]} X$ and we let $\gamma $ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma $ and the post-$\gamma $ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of Levy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

Classification of scaling limits of uniform quadrangulations with a boundary

We study noncompact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the self-similar continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects.

Critical random graphs and the differential equations technique

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t c that specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdős-Renyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n2/3) and (b) the structure of components (rescaled by n−1/3) converge to random fractals related to the continuum random tree. The aim of this note is to give a non-technical overview of recent breakthroughs in this area, emphasizing a particular tool in proving such results called the differential equations technique first developed and used extensively in probabilistic combinatorics in the work of Wormald [52, 53] and developed in the context of critical random graphs by the authors and their collaborators in [10–12].

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.

Statistical Tests for Large Tree-Structured Data

ABSTRACTWe develop a general statistical framework for the analysis and inference of large tree-structured data, with a focus on developing asymptotic goodness-of-fit tests. We first propose a consistent statistical model for binary trees, from which we develop a class of invariant tests. Using the model for binary trees, we then construct tests for general trees by using the distributional properties of the continuum random tree, which arises as the invariant limit for a broad class of models for tree-structured data based on conditioned Galton–Watson processes. The test statistics for the goodness-of-fit tests are simple to compute and are asymptotically distributed as χ2 and F random variables. We illustrate our methods on an important application of detecting tumor heterogeneity in brain cancer. We use a novel approach with tree-based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients. Supplementary materials for this article are available online.

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.

Scaling limits of random Pólya trees

Polya trees are rooted trees considered up to symmetry. We establish the convergence of large uniform random Polya trees with arbitrary degree restrictions to Aldous’ Continuum Random Tree with respect to the Gromov–Hausdorff metric. Our proof is short and elementary, and it is based on a novel decomposition: it shows that the global shape of a random Polya tree is essentially dictated by a large Galton–Watson tree that it contains. We also derive sub-Gaussian tail bounds for both the height and the width, which are optimal up to constant factors in the exponent.

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 (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Wu et al. in Phys Rev Lett 96(14):148702, 2006; Braunstein et al. Int J Bifurc Chaos 17(07):2215–2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006) 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 Lévy processes “without replacement” (Aldous and Limic Electron in J Probab 3(3):59, 1998), 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 a dense collection of hubs (infinite degree vertices), a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal (tau -2)/(tau -3) a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. 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 mathbf {p}-trees is not known, we use general methodology in Athreya et al. (2014) 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.

The harmonic measure of balls in random trees

We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^{\beta}$, where $\beta\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton–Watson trees conditioned to have height greater than $n$. An important ingredient of our approach is the analogous continuous model (related to Aldous’ continuum random tree), where the dimension of harmonic measure of a level set of the tree is equal to $\beta$, whereas the dimension of the level set itself is equal to $1$. The constant $\beta$ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton–Watson trees.

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.