Galton-Watson Trees Research Articles

Hausdorff and packing dimensions of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.

Galton–Watson processes in varying environment and accessibility percolation

This paper deals with branching processes in varying environment with selection, where the offspring distribution depends on the generation and every particle has a random fitness which can only increase along genealogical lineages (descendants with small fitness do not survive). We view the branching process in varying environment (BPVE) as a particular example of branching random walk. We obtain conditions for the survival or extinction of a BPVE (with or without selection), using fixed point techniques for branching random walks. These conditions rely only on the first and second moments of the offspring distributions. Our results can be interpreted in terms of accessibility percolation on Galton-Watson trees. In particular, we obtain that there is no accessibility percolation on almost every Galton-Watson tree where the expected number of offspring grows sublinearly in time, while superlinear growths allows percolation. This result is in agreement with what was found for deterministic trees in Nowak and Krug (Europhysics Letters 101 (2013) 66004).

On the cover time of [formula omitted]-biased walk on supercritical Galton–Watson trees

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 et al. (2018) for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.

Dynamical Pruning of Rooted Trees with Applications to 1-D Ballistic Annihilation

We introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The pruning removes parts of a tree T, starting from the leaves, according to a pruning function defined on descendant subtrees within T. We prove the invariance of critical binary Galton–Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. These results facilitate analysis of the continuum 1-D ballistic annihilation model $$A+A \rightarrow \varnothing $$ for a constant particle density and initial velocity that alternates between the values of $$\pm 1$$ . We show that the model’s shock wave is isometric to the level set tree of the potential function, and the model evolution is equivalent to the generalized dynamical pruning of the shock wave tree.

The coalescent structure of continuous-time Galton–Watson trees

Take a continuous-time Galton–Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+\mu/T$ for some $\mu\in\mathbb{R}$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman’s coalescent, but the times of coalescence have an interesting and highly nontrivial structure. The randomly fluctuating population size, as opposed to constant size populations where the Kingman coalescent more usually arises, have a pronounced effect on both the results and the method of proof required. We give explicit formulas for the distribution of the coalescent times, as well as a construction of the genealogical tree involving a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth–death processes.

Survival and extinction of epidemics on random graphs with general degree

In this paper we establish the necessary and sufficient criterion for the contact process on Galton–Watson trees (resp., random graphs) to exhibit the phase of extinction (resp., short survival). We prove that the survival threshold $\lambda_{1}$ for a Galton–Watson tree is strictly positive if and only if its offspring distribution $\xi$ has an exponential tail, that is, $\mathbb{E}e^{c\xi}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett (2018). On the random graph with degree distribution $\mu$, we show that if $\mu$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for $n^{1+o(1)}$-time $\mathsf{whp}$ (short survival), while for large enough $\lambda$ it runs over $e^{\Theta(n)}$-time $\mathsf{whp}$ (long survival). When $\mu$ is subexponential, we prove that the contact process $\mathsf{whp}$ displays long survival for any fixed $\lambda>0$.

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.

On the Multifractal Analysis of Branching Random Walk on Galton–Watson Tree with Random Metric

We consider a branching random walk $$S_nX(t)$$ on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set $$E(\alpha )$$ of infinite branches in the boundary of tree endowed with random metric along which the average of $$S_n X(t)/n$$ have a given limit point.

On transience of frogs on Galton–Watson trees

We consider an interacting particle system, known as the frog model, on infinite Galton–Watson trees allowing offspring 0 and 1. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient if there are almost surely only finitely many particles returning to the root. In this paper we prove a 0–1-law for transience of the frog model and show the existence of a transient phase for certain classes of Galton–Watson trees.

The Contact Process on Random Graphs and Galton Watson Trees

The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $\lambda_2=0$ and (ii) when it is geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $\lambda_c(n)$ for "prolonged persistence" on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $\lambda_c(n) \to 0$ we give estimates on the rate of convergence. Physicists tell us that $\lambda_c(n) \sim 1/\Lambda(n)$ where $\Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.

Differentiability of the speed of biased random walks on Galton-Watson trees

We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree is differentiable for $\lambda$ 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.

Analyticity for rapidly determined properties of Poisson Galton–Watson trees

Let $T_{\lambda }$ be a Galton–Watson tree with Poisson($\lambda $) offspring, and let $A$ be a tree property. In this paper, we are concerned with the regularity of the function $\mathbb {P}_{\lambda }(A)\coloneqq \mathbb {P}(T_{\lambda }\models A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $\{A_{k}\}$, depending only on the first $k$ vertices in the breadth first exploration of the tree, with a bound in probability of $\mathbb {P}_{\lambda }(A\triangle A_{k}) \le Ce^{-ck}$ over an interval $I = (\lambda _{0}, \lambda _{1})$, then $\mathbb {P}_{\lambda }(A)$ is real analytic in $\lambda $ for $\lambda \in I$. We also present some applications of our results, particularly to properties that are not expressible in first order logic on trees.

The speed of the tagged particle in the exclusion process on Galton–Watson trees

We study two different versions of the simple exclusion process on augmented Galton-Watson trees, the constant speed model and the varying speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with non-vanishing particle density. Moreover, we assume to have initially a particle in the root, the tagged particle. We show for both models that the tagged particle has a positive linear speed and we give explicit formulas for the speeds.

The contact process on periodic trees

A little over 25 years ago Pemantle [6] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $\lambda _{1}$ and $\lambda _{2}$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is $(n,a_{1},\ldots , a_{k})$ with $\max _{i} a_{i} \le Cn^{1-\delta }$ and $\log (a_{1} \cdots a_{k})/\log n \to b$ as $n\to \infty $. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n} $ where $c=(k-b)/2$. This supports Pemantle’s claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

Vertices with fixed outdegrees in large Galton-Watson trees

We are interested in nodes with fixed outdegrees in large conditioned Galton-Watson trees. We first study the scaling limits of processes coding the evolution of the number of such nodes in different explorations of the tree (lexicographical order and contour order) starting from the root. We give necessary and sufficient conditions for the limiting processes to be centered, thus measuring the linearity defect of the evolution of the number of nodes with fixed outdegrees. This extends results by Labarbe & Marckert in the case of the contour-ordered counting process of leaves in uniform plane trees. Then, we extend results obtained by Janson concerning the asymptotic normality of the number of nodes with fixed outdegrees.

Very fat geometric Galton-Watson trees

Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the nth generation and (an, n ∈ ℕ*) is a deterministic positive sequence. We study the local limit of these trees τn as n →∞ and observe three distinct regimes: if (an, n ∈ ℕ*) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ ℕ*) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

Exponential growth and continuous phase transitions for the contact process on trees

We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $\lambda_1$ for weak survival, and the survival probability $p(\lambda)$ is continuous with respect to the infection rate $\lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $\lambda_1<\lambda_2$, which confirms a conjecture of Stacey's \cite{Stacey}. We also prove that if the contact process survives strongly at $\lambda$ then it survives strongly at a $\lambda'<\lambda$, which implies that the process does not survive strongly at the critical value $\lambda_2$ for strong survival.

Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdös-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs (“decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.

The speed of a biased random walk on a Galton-Watson tree is analytic

We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime. This extends the previous work [12] in which it was shown that the speed is differentiable within the range of bias for which a central limit theorem holds.

On the maximal offspring in a subcritical branching process

We consider a subcritical Galton–Watson tree $\mathsf {T}_{n}^{\Omega }$ conditioned on having $n$ vertices with outdegree in a fixed set $\Omega $. The offspring distribution is assumed to have a regularly varying density such that it lies in the domain of attraction of an $\alpha $-stable law for $1<\alpha \le 2$. Our main results consist of a local limit theorem for the maximal degree of $\mathsf {T}_{n}^{\Omega }$, and various limits describing the global shape of $\mathsf {T}_{n}^{\Omega }$. In particular, we describe the joint behaviour of the fringe subtrees dangling from the vertex with maximal degree. We provide applications of our main results to establish limits of graph parameters, such as the height, the non-maximal vertex outdegrees, and fringe subtree statistics.