Let $\mathcal{T}$ denote a Galton--Watson tree with offspring distribution $ξ$ satisfying $\mathbb{E}(ξ) = 1$, and let $\mathcal{T}_n$ be the Galton--Watson tree conditioned to have exactly $n$ nodes. We show that, under a mild moment condition on $ξ$, the number of occurrences of a fixed rooted plane tree $\mathbf{t}$ as a general subtree in $\mathcal{T}_n$ is asymptotically normal as $n \to \infty$, with both mean and variance linear in $n$. In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on $ξ$ is violated, the conclusion may fail.
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 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.
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 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 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 show joint convergence of the Lukasiewicz path and height process for slightly supercritical Galton-Watson forests. This shows that the height processes for supercritical continuous state branching processes as constructed by Lambert (2002) are the limit under rescaling of their discrete counterparts. Unlike for (sub-)critical Galton-Watson forests, the height process does not encode the entire metric structure of a supercritical Galton-Watson forest. We demonstrate that this result is nonetheless useful, by applying it to the configuration model with an i.i.d. power-law degree sequence in …
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…
Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.
The family tree of a Galton-Watson branching process may contain N-ary subtrees, i.e. subtrees whose vertices have at least N>0 children. For family trees without infinite N-ary subtrees, we study how fast N-ary subtrees of height t disappear as t goes to infinity.
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 establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fracti…
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$. The value of the root is the key quantity of interest in general. In this first study, all node values and function values are in a finite set $S$. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on $S$, and the tree $T_n$ is a random Galton--Watson tree of size $n$.
We study the conditions for families of subtrees to exist with high probability (whp) in a Galton-Walton tree of size $n$. We first give a Poisson approximation of fringe subtree counts, which yields the height of the maximal complete $r$-ary fringe subtree. Then we determine the maximal $K_n$ such that every tree of size at most $K_n$ appears as fringe subtree whp. Finally, we study non-fringe subtree counts and determine the height of the maximal complete $r$-ary non-fringe subtree.
When normal and misère games are played on bi-type binary Galton-Watson trees (with vertices coloured blue or red and each having either no child or precisely $2$ children), with one player allowed to move along monochromatic edges and the other along non-monochromatic edges, the draw probabilities equal $0$ unless every vertex gives birth to one blue and one red child. On bi-type Poisson trees where each vertex gives birth to Poisson$(λ)$ offspring in total, the draw probabilities approach $1$ as $λ\rightarrow \infty$. We study such \emph{novel} versions of normal, misère and escape games on …
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.
We show the existance of the stationary distributions of subcritical multitype Galton-Watson processes without any conditions on the mean matrix of the offspring distributions. Some additional properties of the stationary distribution are also proven.
We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton--Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton--Watson trees with finitely many types.
In this short note we obtain necessary and sufficient conditions for the existence of the moments of the stationary distribution of a subcritical multitype Galton-Watson process with immigration.
We give sufficient conditions on the offspring, the initial and the immigration distributions under which a second-order Galton-Watson process with immigration is regularly varying.
A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigrati…
In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom's theorem.
We provide a simple set of sufficient conditions for the weak convergence of discrete Galton-Watson branching processes with immigration to continuous time and continuous state branching processes with immigration.
We give sufficient conditions on the initial, offspring and immigration distributions under which the distribution of a not necessarily stationary Galton--Watson process with immigration is regularly varying at any fixed time.
We analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.
The Galton-Watson process belongs to standard teaching in probability. The basic theorem says that survival is only possible when the fecondity exceeds 1. The classical proof is essentially analytic, using generating functions and convexity arguments. We give here a more probabilistic proof, in the spirit of Bezuidenhout and Grimmett.
We prove that the speed of a $λ$-biased random walk on a supercritical Galton-Watson tree is differentiable for $λ$ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain $2$-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
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 prove that the speed of $λ$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $λ\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
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.
Multi-type inhomogeneous Galton--Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.
We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $λ$-biased random walk on Galton-Watson trees, for which the invarian…
In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, P…
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 prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.
We study the critical parameter u^{*} of random interlacements percolation (introduced by A.S Sznitman in arXiv:0704.2560) on a Galton-Watson tree conditioned on the non-extinction event. Starting from the previous work of A. Teixeira in arXiv:0907.0316, we show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We also characterize this value as the solution of a certain equation.
We propose a new way to condition random trees, that is, condition random trees to have large maximal out-degree. Under this new conditioning, we show that conditioned critical Galton-Watson trees converge locally to size-biased trees with a unique infinite spine. For the sub-critical case, we obtain local convergence to size-biased trees with a unique infinite node. We also study tail of the maximal out-degree of sub-critical Galton-Watson trees, which is essential for the proof of the local convergence.
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Addi…
Ben Arous, Fribergh and Sidoravicius \cite{GAV2014} proved that speed of biased random walk $RW_λ$ on a Galton-Watson tree without leaves is strictly decreasing for $λ\leq \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"ıdékon \cite{EA2013} improved this result to $λ\leq \frac{1}{2}.$ In this paper, we prove that for the $RW_λ$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $λ\in \left[0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"ıdékon \cite{EA2013}.
Recently, Abraham and Delmas constructed the distributions of super-critical Lévy trees truncated at a fixed height by connecting super-critical Lévy trees to (sub)critical Lévy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.
We prove weak convergence on the Skorokhod space of Galton-Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes [Adv. Appl. Probab., 11(1979), 31--62].
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.
We give an expression of the speed of the biased random walk on a Galton--Watson tree. In the particular case of the simple random walk, we recover the result of Lyons, Pemantle and Peres \cite{LyPePe95}. The proof uses a description of the invariant distribution of the environment seen from the particle.
We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.
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.
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…
In this note we consider both the local maximal out-degree and the global maximal out-degree of Galton-Watson trees. In particular, we show that the tail of any local maximal out-degree and that of the offspring distribution are asymptotically of the same order. However for the global maximal out-degree, this is only true in the subcritical case.
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
In this paper, we study the time required for a λ-biased (λ>1) walk to visit all the vertices of a supercritical Galton-Watson tree up to generation n. Inspired by the extremal landscape approach in [Cortines, Louidor, Saglietti 2018] for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.
We consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\leq p_1 <p_2 \leq 1$, there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$
Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.
In this paper, we consider a critical Galton-Watson branching process with immigration stopped at zero $\mathbf{W}$. Some precise estimation on the generation function of the $n$-th population are obtained, and the tail probability of the life period of $\mathbf{W}$ is studied. Based on above results, two conditional limit theorems for $\mathbf{W}$ are established.
In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.
We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form $P (x)s^x$ where P is a polynomial of degree p and s $\in$ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s $\rightarrow$ 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines.
