Related papers: $\beta$-coalescents and stable Galton-Watson trees

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order $\kappa\in (1,2)$. We prove the convergence of the renormalised height function of the walk towards the continuous-time…

We study two different versions of the simple exclusion process on augmented Galton-Watson trees, the constant speed model and the varying speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with…

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…

We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the L\'{e}vy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate…

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq 0\right\} $, with stability \ index $% \alpha \in (0,2]$…

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…

The paper considers the well-known Galton-Watson stochastic branching process. We are dealing with a non-critical case. In the subcritical case, when the mean of the direct descendants of one particle per generation of the time step is less…

Pruning processes $(\mathcal{F}(\theta),\theta\geq 0)$ have been studied separately for Galton-Watson trees and for L\'evy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem…

Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$.…

We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaym\'e-Galton-Watson trees with stably-decaying offspring tails with an exponent $\alpha$ in (1, 2). We show that these…

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of…

This work introduces a construction of conformal processes that combines the theory of branching processes with chordal Loewner evolution. The main novelty lies in the choice of driving measure for the Loewner evolution: given a finite…

This article shows the asymptotics of distribution and moments of the size $X_n$ of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman $n$-coalescent for $n\to\infty$. The Bolthausen-Sznitman $n$-coalescent is a…

We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha\in (1,2]$. Here the harmonic measure refers…

We study several parameters of a random Bienaym\'e-Galton-Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma^2$. Let $f(s)={\bf E}\{s^\xi\}$ be the generating…

We are interested in the structure of large Bienaym\'e-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $\alpha=1$. In stark contrast to the case $\alpha…

We introduce a new, relatively simple, line-breaking construction of the $\alpha$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian…

We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov…

We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index…

This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments…