Related papers: Cutting down trees with a Markov chainsaw

We study the scaling limits of looptrees associated with Bienaym\'e--Galton--Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a "cycle" whose size is its degree. First, we consider BGW trees whose offspring…

We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…

We revisit the random tree model with nearest-neighbour interaction as described in previous work, enhancing growth. When the underlying free Bienaym\'e-Galton-Watson (BGW) model is sub-critical, we show that the (non-Markov) model with…

We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the…

We work on a Galton--Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$-th level of the tree, as $n$ goes to infinity, by a mostly analytic…

A Galton-Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type…

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and…

It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the…

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use…

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which…

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically.…

We study the local limit in distribution of Bienaym{\'e}-Galton-Watson trees conditioned on having large sub-populations. Assuming a generic and aperiodic condition on the offspring distribution, we prove the existence of a limit given by a…

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 investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cram\'er condition the stationary distribution has a power law tail. We…

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 are interested in the structure of multitype Bienaym\'e-Galton-Watson (BGW) trees conditioned on integer linear combinations of the numbers of vertices of given types. We show that, under regularity assumptions on the offspring…

We consider Galton-Watson trees with ${\rm Bin}(d,p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. For $d=2,3$ and any $1/d\leq p_1 <p_2 \leq 1$, we show that there exists a coupling…

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 genealogy of a sample of $k$ individuals taken uniformly without replacement from a continuous-time multitype Bienaym\'e--Galton--Watson process at fixed times. Our results are quite general, requiring only that the process be…

Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue L\'evy trees. L\"ohr, Voisin and Winter (2015) introduced the space of bi-measure…