Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n \rightarrow \infty$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner. In this work, we continue the above investigation and study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $\mathbf{t}_{n}$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $\alpha \in (1,2]$. Our main results establish that, after rescaling, the fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $\alpha$-stable L\'evy tree of index $\alpha \in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\'evy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.Comment: 30 pages, 5 figure

Branching processes with pairwise interactions

In this manuscript, we are interested on the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individual are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by Gonz\'alez Casanova et al. (2017), and includes the so-called logistic branching process which was studied by Lambert (2005). Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments uses the moment dual of BPI-processes which turns out to be a family of diffusions taking values on $[0,1]$, that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries.Comment: 3 table

Percolating, cutting-down and scaling random trees

Trees are a fundamental notion in graph theory and combinatorics as well as a basic object for data structures, algorithms in computer science, statistical physics and the study of epidemics propagating in a network. In recent years, (random) trees have been the subject of many studies and various probabilistic techniques have been developed to describe their behaviors in different settings. In the first part of this thesis, we consider Bernoulli bond-percolation on large random trees. This means that each edge in the tree is removed with some fixed probability and independently of the other edges, inducing a partition of the set of vertices of the tree into connected clusters. We are interested in the supercritical regime, meaning informally that with high probability, there exists a giant cluster, that is of size comparable to that of the entire tree. We study the fluctuations of the size of such giant component, depending on the characteristics of the underlying tree, for two family of trees: b-ary recursive trees and scale-free trees. The approach relies on the analysis of the behavior of certain branching processes subject to rare neutral mutations. In the second part, we study the procedure of cutting-down a tree. We destroy a large tree by removing its edges one after the other and in uniform random order until all the vertices are isolated. We then introduce a random combinatorial object, the so-called cut-tree, that represents the genealogy of the connected components created during the destruction. We investigate the geometry of this cuttree, that depends of course on the nature of the underlying tree, and its implications on the multiple isolation of vertices. The study relies on the close relationship between the destruction process and Bernoulli bond percolation on trees. In the last part of this thesis, we consider asymptotics of large multitype Galton–Watson trees. They are a natural generalization of the usual Galton-Watson trees and describe the genealogy of a population where individuals are differentiated by types that determine their offspring distribution. During the last years, research related to these trees has been developed in connection with important objects and models of growing relevance in modern probability such as random planar maps and non-crossing partitions, to mention just a few. We are more precisely interested in the asymptotic behavior of a function encoding these trees, the well-known height process. We consider offspring distributions that are critical and belong to the domain of attraction of a stable law. We show that these multitype trees behave asymptotically in a similar way as the monotype ones, and that after proper rescaling, they converge weakly to the same continuous random tree, the so-called stable Lévy tree. This extends the result obtained by Miermont [75] in the case of multitype Galton-Watson trees with finite variance

Comportamiento asintótico del CB-Proceso cerca de la extinción

Esta tesis estudia al proceso de ramificación con espacio de estados y tiempo continuo o C.B. proceso el cual puede ser visto como una versión más general del proceso clásico de Galton - Watson. Este fue introducido por Jirina y estudiado por muchos autores, entre los cuales se incluyen Bingham, Grey, Grimvall, Lamperti, por mencionar algunos. Uno de los objetivos de este trabajo es exponer la intima relación del C.B. proceso con los procesos de Lévy espectralmente positivos, debido a que las trayectorias de estos dos procesos están relacionadas a través de la transformación de Lamperti. En particular se estudia al C.B. proceso α-estable (para α ∈ (1,2]) el cual es el proceso resultante al aplicar la transformación de Lamperti a un proceso de Lévy espectralmente positivo α-estable. A partir de ello se obtendrá como consecuencia que el C.B. proceso α-estable tiene la propiedad de auto-similitud. Lo anterior permite establecer test integrales para obtener leyes de logaritmo iterado para el C.B. proceso cerca de la extinción

The distance profile of rooted and unrooted simply generated trees

It is well known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Holder continuous of any order alpha < 1, and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line (0, infinity). The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest

The distance profile of rooted and unrooted simply generated trees

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 distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is H\"older continuous of any order $\alpha<1$, and that it is a.e. differentiable. We note that it cannot be differentiable at $0$, but leave as open questions whether it is Lipschitz, and whether is continuously differentiable on the half-line $(0,\infty)$. The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest

Largest Clusters for Supercritical Percolation on Split Trees

We consider the model of random trees introduced by Devroye [Devroye, 1999], the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. The approach we develop may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we have also studied the case of complete d-regular trees