Galton-Watson Trees Research Articles

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.

Ferromagnetic Ising measures on large locally tree-like graphs

We consider the ferromagnetic Ising model on a sequence of graphs GnGn converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31–51], under an appropriate “continuity” property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with ++ and −− boundary conditions on that tree. Under the extra assumptions that GnGn are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization is the Ising measure with ++ boundary condition on the limiting tree. The “continuity” property holds except possibly for countable many choices of ββ, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton–Watson trees.

Local limits of Galton–Watson trees conditioned on the number of protected nodes

AbstractWe 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.

LOSSY ASYMPTOTIC EQUIPARTITION PROPERTY FOR HIERARCHICAL DATA STRUCTURES

This paper presents a rate-distortion theory for hierarchical networked data structures modelled as tree-indexed multitype process. To be specific, this paper gives a generalized Asymptotic Equipartition Property (AEP) for the Process. The general methodology of proof of the AEP are process level large deviation principles for suitably defined empirical measures for muiltitype Galton-Watson trees.

Scaling limit of multitype Galton–Watson trees with infinitely many types

Nous introduisons une certaine classe d’arbres de Galton–Watson à deux types avec longueurs d’arêtes. Nous prouvons qu’après une renormalisation adéquate, la fonction de hauteur pondérée d’une forêt de tels arbres converge en loi vers le mouvement brownien réfléchi. Nous déduisons ensuite de ceci, sous des hypothèses raisonnables, un principe d’invariance sur les arbres de Galton–Watson multitypes à ensemble de type dénombrable, étendant ainsi un résultat de G. Miermont sur les arbres de Galton–Watson multitypes à ensemble de types fini.

Erratum to: Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in $${\mathbb{R}^d}$$ R d

The paper contains a gap in the justification of the uniform non degeneracy of the measures $${\mu_\varrho}$$ , $${\varrho\in\mathcal{J}}$$ , asserted in Proposition 2.8.1; we used a 0-1 law argument which cannot hold in general due to a conditioning problem; we provide an alternative argument. In the terminology of branching processes, this argument yields new sufficient conditions for an additive martingale on a Galton-Watson tree in varying environment to be almost surely positive conditionally on non extinction of the tree.

Critical Multi-type Galton–Watson Trees Conditioned to be Large

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 .

Local limits of Markov branching trees and their volume growth

We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural assumption on this sequence of probabilities, when their sizes go to infinity, the trees converge in distribution to an infinite tree which also satisfies the Markov branching property. Furthermore, when this infinite tree has a single path from the root to infinity, we give conditions to ensure its convergence in distribution under appropriate rescaling of its distance and counting measure to a self-similar fragmentation tree with immigration. In particular, this allows us to determine how, in this infinite tree, the volume of the ball of radius R centred at the root asymptotically grows with R. Our unified approach will allow us to develop various new applications, in particular to different models of growing trees and cut-trees, and to recover known results. An illustrative example lies in the study of Galton-Watson trees: the distribution of a critical Galton-Watson tree conditioned on its size converges to that of Kesten's tree when the size grows to infinity. If furthermore, the offspring distribution has finite variance, under adequate rescaling, Kesten's tree converges to Aldous' self-similar CRT and the total size of the R first generations asymptotically behaves like R².

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.

Galton-Watson probability contraction

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $\Sigma $, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $\Sigma $. Let $x_{j}(c)$ denote the true probability of the class $j \in \Sigma $ under $Poisson(c)$ regime, and $\vec{x} (c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $\Gamma $, and a map $\Psi = \Psi _{c}: D \rightarrow D$ such that $\vec{x} (c)$ is a fixed point of $\Psi _{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $\Psi $ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.

A study of large fringe and non-fringe subtrees in conditional Galton-Watson trees

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.

Speed of Vertex-Reinforced Jump Process on Galton–Watson Trees

We give an alternative proof of the fact that the vertex-reinforced jump process on Galton–Watson tree has a phase transition between recurrence and transience as a function of $$c$$ , the initial local time, see Basdevant et al. (Ann Appl Probab 22(4):1728–1743, 2012). Further, applying techniques in Aidekon (Probab Theory Relat Fields 142(3–4):525–559, 2008), we show a phase transition between positive speed and null speed for the associated discrete-time process in the transient regime.

The slow regime of randomly biased walks on trees

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.

Wired cycle-breaking dynamics for uniform spanning forests

We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm. Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.

Diseases transmission in a z-ary tree

Consider a Galton–Watson tree [Formula: see text] of height [Formula: see text], each leaf is either infected by one of [Formula: see text] diseases or not infected at all. In other words, [Formula: see text] at generation [Formula: see text] is infected by the [Formula: see text]th infection with probability [Formula: see text] and sane with probability [Formula: see text]. Moreover, the infections are independently distributed for each leaf. Infections spread along the tree based on deterministic specific rules. We study the limit distribution of the disease of the root of [Formula: see text] as [Formula: see text] goes to infinity. We also study the specific case of a [Formula: see text]-ary tree, and we prove convergence of the distribution of the root node for [Formula: see text].

Inference for conditioned Galton-Watson trees from their Harris path

Tree-structured data naturally appear in various fields, particularly in biology where plants and blood vessels may be described by trees, but also in computer science because XML documents form a tree structure. This paper is devoted to the estimation of the relative scale of ordered trees that share the same layout. The theoretical study is achieved for the stochastic model of conditioned Galton-Watson trees. New estimators are introduced and their consistency is stated. A comparison is made with an existing approach of the literature. A simulation study shows the good behavior of our procedure on finite-sample sizes and from missing or noisy data. An application to the analysis of revisions of Wikipedia articles is also considered through real data.

Scaling limit of the recurrent biased random walk on a Galton–Watson tree

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.

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.

Critical value for the contact process with random recovery rates and edge weights on regular tree

In this paper we are concerned with contact processes with random edge weights on rooted regular trees. We assign i.i.d weights on each edge on the tree and assume that an infected vertex infects its healthy neighbor at rate proportional to the weight on the edge connecting them. Under the annealed measure, we define the critical value \lambda_c as the maximum of the infection rate with which the process will die out and define \lambda_e as the maximum of the infection rate with which the process dies out at exponential rate. We show that these two critical values satisfy an identical limit theorem and give an precise lower bound of \lambda_e. We also study the critical value under the quenched measure. We show that this critical value equals that under the annealed measure or infinity according to a dichotomy criterion. The contact process on a Galton-Watson tree with binomial offspring distribution is a special case of our model.

Range and critical generations of a random walk on Galton–Watson trees

In this paper we consider a random walk in random environment on a tree and focus on the frontier case for the underlying branching potential. We study the range $R_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of the order of $n / \log n$. This result is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^2$, which turn out to be the most visited generations. Our proof which involves a quenched analysis gives a description of the typical environments responsible for the behavior of $R_n$.