Multi-type Galton Watson Trees Research Articles

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.

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.

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Levy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offspring distribution. Our approach relies on a remarkable decomposition for multitype trees into monotype trees introduced by Miermont.

Bivariate Markov chains converging to Lamperti transform Markov additive processes

Motivated by various applications, we describe the scaling limits of bivariate Markov chains (X,J) on Z+×{1,…,κ} where X can be viewed as a position marginal and {1,…,κ} is a set of κ types. The chain starts from an initial value (n,i)∈Z+×{1,…,κ}, with i fixed and n→∞, and typically we will assume that the macroscopic jumps of the marginal X are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that X is non-increasing. We then observe different asymptotic regimes according to whether the rate of type change is proportional to, faster than, or slower than the macroscopic jump rate. In these different situations, we obtain in the scaling limit Lamperti transforms of Markov additive processes, that sometimes reduce to standard positive self-similar Markov processes. As first examples of applications, we study the number of collisions in coalescents in varying environment and the scaling limits of Markov random walks with a barrier. This completes previous results obtained by Haas and Miermont (2011) and Bertoin and Kortchemski (2016) in the monotype setting. In a companion paper, we will use these results as a building block to study the scaling limits of multi-type Markov branching trees, with applications to growing models of random trees and multi-type Galton–Watson trees.

Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

Erdős and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter κ∈(1,∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ∈(2,∞], tight in the case κ=2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ∈(1,2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ∈(2,∞]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.

LARGE DEVIATION RESULTS FOR CRITICAL MULTITYPE GALTON-WATSON TREES

In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All our rate functions are expressed in terms of relative entropies.

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.

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 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.

Conditioned multi-type Galton−Watson trees

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.

Unimodularity for multi-type Galton–Watson trees

Fix $n\in\mathbb{N}$. Let $\mathbf{T}_n$ be the set of rooted trees $(T,o)$ whose vertices are labeled by elements of $\{1,...,n\}$. Let $\nu$ be a strongly connected multi-type Galton-Watson measure. We give necessary and sufficient conditions for the existence of a measure $\mu$ that is reversible for simple random walk on $\mathbf{T}_n$ and has the property that given the labels of the root and its neighbors, the descendant subtrees rooted at the neighbors of the root are independent multi-type Galton-Watson trees with conditional offspring distributions that are the same as the conditional offspring distributions of $\nu$ when the types are $\nu$ are ordered pairs of elements of $[n]$. If the types of $\nu$ are given by the labels of vertices, then we give an explicit description of such $\mu$.

A class of multifractal processes constructed using an embedded branching process

We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton-Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step n, we can generate step n+1 in O(log n) operations. Detailed pseudo-code for this algorithm is provided.

Absolutely Continuous Spectrum for Multi-type Galton Watson Trees

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.

Central limit theorem for biased random walk on multi-type Galton-Watson trees

Let $\mathcal{T}$ be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The $\lambda$-biased random walk $(X_t)_{t\ge0}$ on $\mathcal{T}$ is the nearest-neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $\lambda/(\lambda+d_v)$, and to each of the offspring with probability $1/(\lambda+d_v)$. This walk is recurrent for $\lambda\ge\rho$ and transient for $0\le\lambda<\rho$, with $\rho$ the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. Subject to finite moments of order $p>4$ for the offspring distributions, we prove the following quenched CLT for $\lambda$-biased random walk at the critical value $\lambda=\rho$: for almost every $\mathcal{T}$, the process $|X_{\lfloor nt \rfloor}|/\sqrt{n}$ converges in law as $n\to\infty$ to a reflected Brownian motion rescaled by an explicit constant. This result was proved under some stronger assumptions by Peres-Zeitouni (2008) for single-type Galton-Watson trees. Following their approach, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (generalizing the measure constructed in the single-type setting by Peres-Zeitouni), and the construction of appropriate harmonic coordinates. In carrying out this program we prove moment and conductance estimates for MGW trees, which may be of independent interest. In addition, we extend our construction of the reversing measure to a biased random walk with random environment (RWRE) on MGW trees, again at a critical value of the bias. We compare this result against a transience-recurrence criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson trees.

Asymptotic equipartition properties for simple hierarchical and networked structures

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large $n$, a networked data structure consisting of $n$ units connected by an average number of links of order $n/log n$ can be coded by about $nH$ bits, where $H$ is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures.

Invariance principles for spatial multitype Galton–Watson trees

Nous montrons que les arbres de Galton–Watson multitypes, dont les lois de reproduction sont irrductibles et de matrices de covariance finies, admettent pour limite d’chelle l’arbre continu brownien. La clef de notre tude est une dcomposition ancestrale pour les arbres multitypes marqus, et une mthode par rcurrence sur le nombre de types. Nous couplons ensuite la structure gnalogique avec des dplacements spaciaux, dont la loi de saut peut dpendre localement de la structure de l’arbre, et nous montrons que les arbres spatiaux obtenus convergent vers le serpent brownien, sous certaines hypothses de moments.

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton–Watson trees. As a consequence, we find that the expected volume of the ball of radius r around a marked point in the limit random surface is Θ(r4).

Large deviations of Markov chains indexed by random trees

Given a finite typed rooted tree T with n vertices, the empirical subtree measure is the uniform measure on the n typed subtrees of T formed by taking all descendants of a single vertex. We prove a large deviation principle in n, with explicit rate function, for the empirical subtree measures of multitype Galton–Watson trees conditioned to have exactly n vertices. In the process, we extend the notions of shift-invariance and specific relative entropy—as typically understood for Markov fields on deterministic graphs such as Z d —to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with n vertices.