Related papers: Total progeny in almost critical multi-type Galton…
The classical Galton--Watson process works with a fixed probability of fission at each time step. One of the generalizations is that the probabilities depend on time. We consider one of the most complex and interesting cases when we do not…
This article is an essay, both expository and argumentative, on the Galton-Watson process as a tool in the domain of Branching Processes. It is at the same time the author's ways to honour two distinguished scientists in this domain, both…
A Galton-Watson process in a varying environment is a discrete time branching process where the offspring distributions vary among generations. It is known that in the critical case, these processes have a Yaglom limit, that is, a suitable…
We employ the framework of multitype Galton-Watson processes to model a population of dividing cells. The cellular type is represented by its biological age, defined as the count of harmful proteins hosted by the cell. The stochastic…
We consider a subcritical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We consider the event $% \mathcal{A}_{i}(n)$ that all individuals alive at time $n$ are offspring of the…
We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when…
We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour, which started from the root of a tree, spreads out through an infinite…
The paper studies a class of critical Markov branching processes with infinite variance of the offspring distribution. The processes admit also an immigration component at the jump-points of a non-homogeneous Poisson process, assuming that…
In this paper we are interested in a random walk in a random environment on a super-critical Galton-Watson tree. We focus on the recurrent cases already studied by Y. Hu and Z. Shi and G. Faraud. We prove that the largest generation…
The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade)…
We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We…
There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities' we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed picture of…
We provide a simple set of sufficient conditions for the weak convergence of discrete Galton-Watson branching processes with immigration to continuous time and continuous state branching processes with immigration.
Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical…
Let $A$ be a primitive matrix and let $\lambda$ be its Perron-Frobenius eigenvalue. We give formulas expressing the associated normalized Perron-Frobenius eigenvector as a simple functional of a multitype Galton-Watson process whose mean…
The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size $n$…
We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite…
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 critical percolation on a supercritical Galton-Watson tree. We show that, when the offspring distribution is in the domain of attraction of an $\alpha$-stable law for some $\alpha \in (1,2)$, or has finite variance, several…
We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton-Watson…