Watson Trees Research Articles

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.

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.

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.

Tree limits and limits of random trees

AbstractWe explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

Directed Polymers on Infinite Graphs

We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the $\mathbb Z^d$/SRW result.

Metric dimension of critical Galton–Watson trees and linear preferential attachment trees

The metric dimension of a graph G is the minimal size of a subset R of vertices of G that, upon reporting their graph distance from a distinguished (source) vertex v⋆, enable unique identification of the source vertex v⋆ among all possible vertices of G. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton–Watson trees conditioned to have size n, and growing general linear preferential attachment trees. The former class includes uniform random trees, the latter class includes Yule-trees (also called random recursive trees), m-ary increasing trees, binary search trees, and positive linear preferential attachment trees. In all these cases, we are able to identify the limiting constant in the LLN explicitly. Our result relies on the insight that the metric dimension can be related to subtree properties, and hence we can make use of the powerful fringe-tree literature developed by Aldous and Janson et al.

Local Convergence of Critical Random Trees and Continuous-State Branching Processes

We study the local convergence of critical Galton–Watson trees and Lévy trees under various conditionings. Assuming a very general monotonicity property on the measurable functions of critical random trees, we show that random trees conditioned to have large function values always converge locally to immortal trees. We also derive a very general ratio limit property for measurable functions of critical random trees satisfying the monotonicity property. Finally we study the local convergence of critical continuous-state branching processes, and prove a similar result.

Rerooting Multi-type Branching Trees: The Infinite Spine Case

We prove local convergence results for rerooted conditioned multi-type Galton–Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.

The TASEP on Galton–Watson trees

We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from x to y at rate rx,y provided y is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time t converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first n particles decouple.

The Horton–Strahler number of conditioned Galton–Watson trees

The Horton–Strahler number of a tree is a measure of its branching complexity. It is also known in the literature as the register function. We show that for critical Galton–Watson trees with finite variance, conditioned to be of size n, the Horton–Strahler number grows as 1 2log2n in probability. We further define some generalizations of this number. Among these are the rigid Horton–Strahler number and the k-ary register function, for which we prove asymptotic results analogous to the standard case.

On general subtrees of a conditioned Galton–Watson tree

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.

K-cut model for the Brownian continuum random tree

To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the k-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the k-cut model to isolate the root of a Galton–Watson tree with a finite-variance offspring law and conditioned to have n nodes, when divided by n1−1∕2k, converges in distribution to some random variable defined on the Brownian CRT. We provide here a direct construction of the limit random variable, relying upon the Aldous–Pitman fragmentation process and a deterministic time change.

Hausdorff dimension of the uniform measure of Galton–Watson trees without the XlogX condition

Nous considérons un arbre de Galton–Watson dont le nombre d’enfants $\nu $ a une moyenne finie. La mesure uniforme sur la frontière de l’arbre s’obtient en chargeant chaque sommet de la $n$-ième génération avec une masse $1$, puis en prenant la limite $n\to \infty $. Dans le cas $E[\nu \log (\nu )]<\infty $, cette mesure est bien étudiée, et l’on sait que la dimension de Hausdorff de la mesure est égale à $\log (m)$ (J. Lond. Math. Soc. (2) 24 (1981) 373–384; Ergodic Theory Dynam. Systems 15 (1995) 593–619). Lorsque $E[\nu \log (\nu )]=\infty $, nous montrons que la dimension est $0$. Cela répond à une question posée par Lyons, Pemantle et Peres (In Classical and Modern Branching Processes. Proceedings of the IMA Workshop (1997) 223–237 Springer).

Renewal in Hawkes processes with self-excitation and inhibition

AbstractWe investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

The boundary of random planar maps via looptrees

We study the scaling limits of looptrees associated with Bienaymé–Galton–Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a “cycle” whose size is its degree. First, we consider BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution. We prove that the Brownian CRT is the scaling limit of the associated looptrees, thereby confirming a prediction of [18]. Then, we deal with BGW trees whose offspring distribution is critical and heavy-tailed. We show that the scaling limit of the associated looptrees is a multiple of the unit circle. This corresponds to a so-called condensation phenomenon, meaning that the underlying tree exhibits a vertex with macroscopic degree. Here, we rely on an invariance principle for random walks with negative drift, which is of independent interest. Finally, we apply these results to the study of the scaling limits of large faces of Boltzmann planar maps. We complete the results of [50] and establish a phase transition for the topology of these maps in the non-generic critical regime.

Quenched Survival of Bernoulli Percolation on Galton–Watson Trees

We explore the survival function for percolation on Galton–Watson trees. Letting g(T, p) represent the probability a tree T survives Bernoulli percolation with parameter p, we establish several results about the behavior of the random function $$g(\mathbf{T}, \cdot )$$ , where $$\mathbf{T}$$ is drawn from the Galton–Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $$k\text {th}$$ -order Taylor expansion of $$g(\mathbf{T}, \cdot )$$ at criticality in terms of limits of martingales defined from $$\mathbf{T}$$ (this requires a moment condition depending on k); and a proof that the $$k\text {th}$$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton–Watson tree.