Galton-Watson Trees Research Articles

Capacity of the Range of Branching Random Walks in Low Dimensions

Consider a branching random walk $(V_u)_{u\in \mathcal T^{IGW}}$ in $\mathbb Z^d$ with the genealogy tree $\mathcal T^{IGW}$ formed by a sequence of i.i.d. critical Galton-Watson trees. Let $R_n $ be the set of points in $\mathbb Z^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in $\mathcal T^{IGW}$. Our main result states that for $d\in \{3, 4, 5\}$, the capacity of $R_n$ is almost surely equal to $n^{\frac{d-2}{2}+o(1)}$ as $n \to \infty$.

Root estimation in Galton–Watson trees

AbstractGiven only the free‐tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum‐likelihood estimator for the root of a free tree when the underlying tree is a size‐conditioned Galton–Watson tree and calculate its probability of being correct.

Scaling limit of the subdiffusive random walk on a Galton–Watson tree in random environment

We consider a random walk on a Galton–Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the real tree coded by the latter continuous-time height process.

Universal height and width bounds for random trees

We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson [13], and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman [9] in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees. Finally, we propose and justify a change to the conventions of branching process nomenclature: the name “Galton–Watson trees” should be permanently retired by the community, and replaced with the name “Bienaymé trees”.

Critical window for the vacant set left by random walk on the configuration model

We study the simple random walk on the configuration model with given degree sequence $(d_1^n, \dots ,d_n^n)$ and investigate the connected components of its vacant set at level $u>0$. We show that the size of the maximal connected component exhibits a phase transition at level $u^*$ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size $n^{-1/3}$ around $u^*$ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Renyi random graph.

Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (i.e. the set of finite words written in the alphabet {1,…,b}, with b≥2). The BRW is indexed by a critical Galton–Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, γ∈(1,2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 1∕2 to move closer to the root of Wb and probability 1∕(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)n∈N satisfying an→∞ such that the metric space (Rb(n),an−1dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

The sum of powers of subtree sizes for conditioned Galton–Watson trees

We study the additive functional Xn(α) on conditioned Galton–Watson trees given, for arbitrary complex α, by summing the αth 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 Xn(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 extensions of, limiting distribution results for α in various regions of the complex plane. We focus mainly on the case where Reα>0, for which Xn(α), suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution ξ of the conditioned Galton–Watson tree, assuming only that Eξ=1 and 0<Varξ<∞. Under a weak extra moment assumption on ξ, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when Reα>12, the limit random variable Y(α) can be expressed as a function of a normalized Brownian excursion.

Sharpness of the phase transition for parking on random trees

AbstractRecently, a phase transition phenomenon has been established for parking on random trees. We extend the results of Curien and Hénard on general Bienaymé–Galton–Watson trees and allow different car arrival distributions depending on the vertex outdegrees. We then prove that this phase transition is sharp by establishing a large deviations result for the flux of exiting cars. This has consequences on the offcritical geometry of clusters of parked spots which displays similarities with the classical Erdős–Renyi random graph model.

Branching Random Walks Conditioned on Particle Numbers

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.

Exponential rate for the contact process extinction time

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.

Frozen percolation on the binary tree is nonendogenous

In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time an edge opens provided neither of its end vertices is part of an infinite open cluster; in the opposite case it freezes. Aldous (Math. Proc. Cambridge Philos. Soc. 128 (2000) 465–477) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (Ann. Appl. Probab. 15 (2005) 1047–1110), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton–Watson tree that has nice scale invariant properties.

Non-Gaussian fluctuations of randomly trapped random walks

AbstractIn this paper we consider the one-dimensional, biased, randomly trapped random walk with infinite-variance trapping times. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton–Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

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 $\alpha&lt;1$ , 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.

Random fractals and their intersection with winning sets

AbstractWe show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$.We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

Continuous phase transitions on Galton–Watson trees

AbstractDistinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

Scaling limits of multi-type Markov Branching trees

We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton–Watson fashion, with the rule that the size of any individual is at least the sum of the sizes of its children. Assuming that the macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased child is proportional to the probability of macroscopic splitting (the critical regime, in which we get in the limit multi-type fragmentation trees with indeed several types), smaller than the probability of macroscopic splitting (the solo regime, in which the limit trees are monotype as we never see a type change), or larger than the probability of macroscopic splitting (the mixing regime, in which case the types mix in the limit and we get monotype fragmentation trees). This framework allows us to unify models which may a priori seem quite different, a strength which we illustrate with two notable applications. The first one concerns the description of the scaling limits of growing models of random trees built by gluing at each step on the current structure a finite tree picked randomly in a finite alphabet of trees, extending Rémy’s well-known algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical regime with multi-type fragmentation trees in the scaling limit, or in the solo regime. The second application concerns the scaling limits of large multi-type critical Galton–Watson trees when the offspring distributions all have finite second moments. This topic has already been studied but our approach gives a different proof and we improve on previous results by relaxing some hypotheses. We are then in the mixing regime: the scaling limits are always multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework.

Galton–Watson games

We address two‐player combinatorial games whose graph of positions is a directed Galton–Watson tree. We consider normal and misère rules (where a player who cannot move loses or wins, respectively), as well as an “escape game” in which one designated player loses if either player cannot move. We study phase transitions for the probability of a draw or escape under optimal play, as the offspring distribution varies. Across a range of natural cases, we find that the transitions are continuous for the normal and misère games but discontinuous for the escape game; we also exhibit examples where these properties fail to hold. We connect the nature of the phase transitions to the length of the game under optimal play. We establish inequalities between the different games. For instance, the draw probability is no smaller in the misère game than in the normal game.

Invariance and attraction properties of Galton–Watson trees

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 {qk} that has a power tail qk∼Ck−(1+1/q0), where q0∈(1/2,1). Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the tail behavior of the initial Galton–Watson offspring distribution. The invariant measures satisfy the Toeplitz property for the Tokunaga coefficients and obey the Horton law with exponent R=(1−q0)−1/q0.

Emergence of extended states at zero in the spectrum of sparse random graphs

We confirm the long-standing prediction that c=e≈2.718 is the threshold for the emergence of a nonvanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős–Rényi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A−z)−1 near the singular point z=0, where A is the adjacency operator of the Poisson–Galton–Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton–Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.

Age evolution in the mean field forest fire model via multitype branching processes

We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős–Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton–Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.