Predator–Prey Dynamics on Infinite Trees: A Branching Random Walk Approach

We are interested in predator–prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase–escape process, introduced by Kordzakhia with a variant by Bordenave who sees it as a rumor propagation model, and the birth-and-assassination process, introduced by Aldous and Krebs. We exhibit a coupling between these processes and branching random walks killed at the origin. This sheds new light on the chase–escape and birth-and-assassination processes, which allows us to recover by probabilistic means previously known results and also to obtain new results. For instance, we find the asymptotic behavior of the tail of the number of infected individuals in both the subcritical and critical regimes for the chase–escape process and show that the birth-and-assassination process ends almost surely at criticality.     

Critical Multi-type Galton–Watson Trees Conditioned to be Large

Under minimal condition, we prove the local convergence of a critical multi-type Galton–Watson tree conditioned on having a large total progeny by types toward a multi-type Kesten’s tree. We obtain the result by generalizing Neveu’s strong ratio limit theorem for aperiodic random walks on \(\mathbb {Z}^d\).     

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.     

Conditioning Galton–Watson Trees on Large Maximal Outdegree

We propose a new way to condition random trees, that is, conditioning random trees to have large maximal outdegree. Under this conditioning, we show that conditioned critical Galton–Watson trees converge locally to size-biased trees with a unique infinite spine. For the subcritical case, we obtain the local convergence to size-biased trees with a unique infinite node. We also study the tail of the maximal outdegree of subcritical Galton–Watson trees, which is essential for the proof of the local convergence.     

Stochastic Ordering of Infinite Geometric Galton–Watson Trees

We consider Galton–Watson trees with Geom\((p)\) offspring distribution. We let \(T_{\infty }(p)\) denote such a tree conditioned on being infinite. We prove that for any \(1/2\le p_1 <p_2 \le 1\), there exists a coupling between \(T_{\infty }(p_1)\) and \(T_{\infty }(p_2)\) such that \({\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1\).     

Speed of Vertex-Reinforced Jump Process on Galton–Watson Trees

We give an alternative proof of the fact that the vertex-reinforced jump process on Galton–Watson tree has a phase transition between recurrence and transience as a function of \(c\), the initial local time, see Basdevant et al. (Ann Appl Probab 22(4):1728–1743, 2012). Further, applying techniques in Aidékon (Probab Theory Relat Fields 142(3–4):525–559, 2008), we show a phase transition between positive speed and null speed for the associated discrete-time process in the transient regime.     

Biased random walk on critical Galton–Watson trees conditioned to survive

We consider the biased random walk on a critical Galton–Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.     

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

We study the typical behavior of the harmonic measure in large critical Galton–Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index \(\alpha \in (1,2]\). Let \(\mu _n\) denote the hitting distribution of height n by simple random walk on the critical Galton–Watson tree conditioned on non-extinction at generation n. We extend the results of Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584, 2015) to prove that, with high probability, the mass of the harmonic measure \(\mu _n\) carried by a random vertex uniformly chosen from height n is approximately equal to \(n^{-\lambda _\alpha }\), where the constant \(\lambda _\alpha >\frac{1}{\alpha -1}\) depends only on the index \(\alpha \). In the analogous continuous model, this constant \(\lambda _\alpha \) turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for \(\lambda _\alpha \), we are able to show that \(\lambda _\alpha \) decreases with respect to \(\alpha \in (1,2]\), and it goes to infinity at the same speed as \((\alpha -1)^{-2}\) when \(\alpha \) approaches 1.     

Approximation of SDEs by Population-Size-Dependent Galton–Watson Processes

A certain class of stochastic differential equations, containing the Cox–Ingersoll–Ross model and the geometric Brownian motion, is considered. The corresponding solutions are approximated weakly by discrete-time population-size-dependent Galton–Watson processes with immigration. The long-time behavior of the limiting processes is also investigated.     

Detection of cellular aging in a Galton–Watson process

We consider the bifurcating Markov chain model introduced by Guyon to detect cellular aging from cell lineage. To take into account the possibility for a cell to die, we use an underlying super-critical binary Galton–Watson process to describe the evolution of the cell lineage. We give in this more general framework a weak law of large number, an invariance principle and thus fluctuation results for the average over all individuals in a given generation, or up to a given generation. We also prove that the fluctuations over each generation are independent. Then we present the natural modifications of the tests given by Guyon in cellular aging detection within the particular case of the auto-regressive model.     

KKM—A Topological Approach For Trees

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems.A 2-tree hypergraph is a family of nonempty subsets of T R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets.For each such hypergraph H we denote by (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by (H) the maximal number of disjoint sets in it.In this paper we prove that in a 2-tree hypergraph (H)2(H) and in a homogeneous 2-tree hypergraph (H)3(H). This improves the result of Alon [3], that (H)8(H) in both cases.Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.     

Predicting extinction or explosion in a Galton–Watson branching process

Based on observations $X_1,\dots ,X_n$ X 1 , … , X n of successive generations of a discrete-parameter Galton–Watson branching process, one wishes to predict whether extinction or explosion will ultimately occur. This problem can be formulated as a simple hypothesis-testing problem to which the Neyman–Pearson Lemma is directly applicable if the extinction probability is known or estimable. If it is not, valid (but conservative) tests still can be obtained.     

Functional limit theorems for Galton–Watson processes with very active immigration

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979).     

Transient random walks in random environment on a Galton–Watson tree

We consider a transient random walk (X _n ) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X _n is of order of magnitude n ^Λ, with ${\Lambda \in (0,1)}$ . We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81–101, 2006).     

Statistical inference of 2-type critical Galton–Watson processes with immigration

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton–Watson branching process with immigration is described. We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well.     

On Aggregation of Subcritical Galton–Watson Branching Processes with Regularly Varying Immigration

Lithuanian Mathematical Journal - Abstract. We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton–Watson branching...     

Limit Theorems for Critical Galton–Watson Processes with Immigration Stopped at Zero

In this paper, we consider a critical Galton–Watson branching process with immigration stopped at zero W. Some precise estimation on the probability generating function of the n-th population are obtained, and the tail probability of the life period of W is studied. Based on above results,two conditional limit theorems for W are established.