We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly $n$ vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the increments of the spatial positions along edges of the tree are independent variables distributed according to a symmetric probability distribution on the real line. We then condition on the event that all spatial positions are nonnegative. Under suitable assumptions on the offspring distribution and the spatial displacements, we prove that these conditioned spatial trees converge as $n\to\infty$, modulo an appropriate rescaling, towards the conditioned Brownian tree that was studied in previous work. Applications are given to asymptotics for random quadrangulations.
Let $S(n)$ be a centered random walk with finite second moment. We consider the integrated random walk $T(n) = S(0)+S(1)+\dots+S(n)$. We prove invariance principles for the meander and for the bridge of this process, under the condition that the integrated random walk remains positive. Furthermore, we prove the functional convergence of its Doob's $h$-transform to the $h$-transform of the Kolmogorov diffusion conditioned to stay positive.
We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and W…
Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line …
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.
We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.
Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .
We investigate conditioning Galton-Watson trees on general recursive-type events, such as the event that the tree survives until a specific level. It turns out that the conditioned tree is again a type of Galton-Watson tree, with different types of offspring and a level-dependent offspring distribution, which will all be given explicitly. As an interesting application of these results, we will calculate the expected cost of searching a tree until reaching a given level.
We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.
We give an invariance principle for very general additive functionals of conditioned Bienaym{é}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{é}vy tree. This includes the case when the offspring distribution has finite variance (the L{é}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.
We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.
We provide a complete picture of the local convergence of critical or subcritical Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set. The generic case, where the limit is a random tree with an infinite spine has been treated in a previous paper. We focus here on the non-generic case, where the limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.
We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Levy trees. This extends previous results due to several authors.
We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton-Watson tree having $n$ leaves. Secondly, we let $t_n$ be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly $n$ leaves. We show that the rescaled Lukasiewicz path and contour function of $t_n$ converge respectively to $X^{exc}$ and $H^{exc}$, …
We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaymé-Galton-Watson trees conditioned on having a large fixed size, whose offspring distribution belongs to the domain of attraction of a stable law. This extends results obtained for the height and width by Addario-Berry, Devroye & Janson in the finite variance case.
We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit of finite variance Galton-Watson trees condition on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to this is a generalization of the classical Otter-Dwass formula.
We investigate a neutral model for speciation and extinction, the constant rate birth-death process. The process is conditioned to have $n$ extant species today, we look at the tree distribution of the reconstructed trees-- i.e. the trees without the extinct species. Whereas the tree shape distribution is well-known and actually the same as under the pure birth process, no analytic results for the speciation times were known. We provide the distribution for the speciation times and calculate the expectations analytically. This characterizes the reconstructed trees completely. We will show how …
We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the so-called Integrated Super-Brownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive half-line. …
We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.
We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton--Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton--Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton--Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The proofs use a well-known connection to a random a…
Drmota and Gittenberger (1997) proved a conjecture due to Aldous (1991) on the height profile of a Galton-Watson tree with an offspring distribution of finite variance, conditioned on a total size of $n$ individuals. The conjecture states that in distribution its shape, more precisely its scaled height profile coincides asymptotically with the local time process of a Brownian excursion of duration 1. We give a proof of the result, which extends to the case of an infinite variance offspring distribution. This requires a different strategy, since in the infinite variance case there is no longer …
We show that the number of copies of a given rooted tree in a conditioned Galton-Watson tree satisfies a law of large numbers under a minimal moment condition on the offspring distribution.
We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some suitable moment assumptions.
In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergenc…
In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $α\in (1, 2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced \cite{LGLJ1, LGLJ1}. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one…
An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, as has been shown in a special case by Luczak and Winkler.
We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index $α\in(1,2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $2α/(2α-1)$. Furthermore, we demonstrate that when $α\in(1,2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $α=2$. In the course…
We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton-Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index $θ\in (1,2]$, conditioned on having total progeny $n$, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive Lévy process of index $θ$. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of h…
In this paper, we show that a Galton-Watson tree conditioned to have a fixed number of particles in generation $n$ converges in distribution as $n\rightarrow\infty$, and with this tool we study the span and gap statistics of a branching random walk on such trees, which is the discrete version of Ramola, Majumdar and Schehr 2015, generalized to arbitrary offspring and displacement distributions with moment constraints.
We study $S(\mathcal T_{n})$, the number of subtrees in a conditioned Galton--Watson tree of size $n$. With two very different methods, we show that $\log(S(\mathcal T_{n}))$ has a Central Limit Law and that the moments of $S(\mathcal T_{n})$ are of exponential scale.
We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed.
We prove invariance principles for a mulditimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of $h$-transformed random walk to the corresponding $h$-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.
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 consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to th…
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 a particular type of subcritical Galton--Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical ca…
We study the maximal degree of (sub)critical L{é}vy trees which arise as the scaling limits of Bienaym{é}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{é}vy tree conditioned to have a fixed maximal degree. In the case where the L{é}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{é}vy tree can be obtained by grafting a L{é…
We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under the conditions of Möhle and Sagitov, this convergence holds for all tree-valued Fleming-Viot processes under consideration in the dust-free case, and for the Fleming-Viot processes with values in the space of distance matrix distributions in the case with dust. Convergence to Fleming-Viot processes with values in the space of marked metric measure spaces in the case with dust is ensured by an additio…
We study the diameter of L{é}vy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L{é}vy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of L{é}vy trees and we prove that it is realized by a unique pair of points. We prove that the law of L{é}vy trees conditioned to have a fixed diameter r $\in$ (0, $\infty$) is obtained by glueing at their respective roots two independent size-biased L{é}vy trees …
We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution towards Aldous' Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton-Watson tree structure.
We consider branching random walks built on Galton-Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of "globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, "globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the "lineage'' of the nodes in a Galton-Watson tree conditioned b…
We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned …
For a complex number $α$, we consider the sum of the $α$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(α)$ have been determined for $\Reα\neq 0$, revealing a transition between a complex normal limiting distribution for $\Reα< 0$ and a non-normal limiting distribution for $\Reα> 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Reα= 0$. The same results are also established in the case of the so-called shape functional $X_n'(…
We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, agrees up to generation $K$ with a regular $μ$-ary tree, where $μ$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $\log(1/\eps)$. More precisely, we show that if $μ\ge 2$ then with high probability as $\eps \downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $μ$-ary tree, providing an ex…
We study the additive functional $X_n(α)$ on conditioned Galton-Watson trees given, for arbitrary complex $α$, by summing the $α$th power of all subtree sizes. Allowing complex $α$ is advantageous, even for the study of real $α$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Reα< 0$, we prove that $X_n(α)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $α$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence e…
Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes - the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912). We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the …
We consider stochastic processes on complete, locally compact tree-like metric spaces $(T,r)$ on their "natural scale" with boundedly finite speed measure $ν$. Given a triple $(T,r,ν)$ such a speed-$ν$ motion on $(T,r)$ can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all $x,y\in T$ and all positive, bounded measurable $f$, \[ \mathbb{E}^x [ \int^{τ_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_Tν(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] where $c(x,y,z)$ denotes the branch point generated by $x,y,z$. If $(T,r)$ is a discrete tree, $X$ is …
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 the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.
