Related papers: Conditioned Galton-Watson trees do not grow
We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results…
We show that it is impossible to improve the high-energy behavior of the tree-level four-point amplitude of a massive spin-2 particle by including the exchange of any number of scalars and vectors in four spacetime dimensions. This…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…
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…
We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on the bias $\beta$, such that $X_n$ is of…
We study the conditions for families of subtrees to exist with high probability (whp) in a Galton-Walton tree of size $n$. We first give a Poisson approximation of fringe subtree counts, which yields the height of the maximal complete…
We prove asymptotic normality for the number of fringe subtrees isomorphic to any given tree in uniformly random trees with given vertex degrees. As applications, we also prove corresponding results for random labelled trees with given…
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…
In this note we consider both the local maximal out-degree and the global maximal out-degree of Galton-Watson trees. In particular, we show that the tail of any local maximal out-degree and that of the offspring distribution are…
We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau$n distributed as $\tau$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We…
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random…
Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when…
We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have…
We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine…
We give an expression of the speed of the biased random walk on a Galton--Watson tree. In the particular case of the simple random walk, we recover the result of Lyons, Pemantle and Peres \cite{LyPePe95}. The proof uses a description of the…
We study the typical behavior of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution has finite variance. The harmonic measure considered here refers to the hitting distribution of height $n$ by…
Motivated by the study of pattern avoidance in the context of permutations and ordered partitions, we consider the enumeration of weak-ordering chains obtained as leaves of certain restricted rooted trees. A tree of order $n$ is generated…
Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution…
We will give a criterion for the amenability of arbitrary locally finite trees. The criterion is based on the trimming operator which is defined on the space of trees. As an application, we obtain a necessary and sufficient condition for…
We provide a fundamental result for bucket increasing trees, which gives a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent…