Related papers: Constructing and searching conditioned Galton-Wats…
We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on itsout-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level n -- 1…
We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for…
We study a particular type of subcritical Galton--Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large…
In this paper we consider inhomogeneous Galton-Watson trees, and derive various moments for such processes: the number of vertices, the number of leaves, and the height of the tree. Also we make a simple condition of finiteness. We use…
We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore…
The aim of this lecture is to give an overview of old and new resultson Bienaym\'e-Galton-Watson (BGW) trees. After introducing the framework of discretetrees, we first give alternative proofs of classical results on theextinction…
In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such…
We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the…
We give a realization of the stable L\'evy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process $\{{\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous…
We study the heavy path decomposition of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is…
We provide a complete picture of the local convergence of critical or subcritical Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set. The generic case, where the limit is a random tree with…
We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly $n$ vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the…
The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of…
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…
We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on its outdegree. We give a complete picture of the local convergence of critical or sub-critical marked Galton-Watson…
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…
We consider a model of random loops on Galton-Watson trees with an offspring distribution with high expectation. We give the configurations a weighting of $\theta^{\#\text{loops}}$. For many $\theta>1$ these models are equivalent to certain…
We consider the set of random Bienaym\'e-Galton-Watson trees with a bounded number of offspring and bounded number of generations as a statistical mechanics model: a random tree is a rooted subtree of the maximal tree; the spin at a given…