Related papers: Inversions in split trees and conditional Galton--…
We propose a new way to condition random trees, that is, condition random trees to have large maximal out-degree. Under this new conditioning, we show that conditioned critical Galton-Watson trees converge locally to size-biased trees with…
We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton-Watson tree conditioned to be large…
Let $T\_n$ denote the set of unrooted labeled trees of size $n$ and let $T\_n$ be a particular (finite, unlabeled) tree. Assuming that every tree of $T\_n$ is equally likely, it is shown that the limiting distribution as $n$ goes to…
We consider a fragmentation of discrete trees where the internal vertices are deleted independently at a rate proportional to their degree. Informally, the associated cut-tree represents the genealogy of the nested connected components…
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…
The $k$-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges…
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…
We 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…
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian…
We study the additive functional $X_n(\alpha)$ on conditioned Galton-Watson trees given, for arbitrary complex $\alpha$, by summing the $\alpha$th power of all subtree sizes. Allowing complex $\alpha$ is advantageous, even for the study of…
Consider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this…
Distinguishing 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…
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 are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random;…
We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit…
Consider a random walk in random environment on a supercritical Galton--Watson tree, and let $\tau_n$ be the hitting time of generation $n$. The paper presents a large deviation principle for $\tau_n/n$, both in quenched and annealed cases.…
Out-Of-Distribution (OOD) generalization is an essential topic in machine learning. However, recent research is only focusing on the corresponding methods for neural networks. This paper introduces a novel and effective solution for OOD…
We consider a branching population where individuals have i.i.d.\ life lengths (not necessarily exponential) and constant birth rate. We let $N_t$ denote the population size at time $t$. %(called homogeneous, binary Crump--Mode--Jagers…
We study the limiting behavior of a Bienayme-Galton-Watson tree conditioned to have a large number of vertices and either a fixed number of leaves or a fixed number of internal nodes. The first biconditioning gives a universal result with…
We study random trees which are invariant in law under the operation of contracting each edge independently with probability $p\in(0,1)$. We show that all such trees can be constructed through Poissonian sampling from a certain class of…