Percolating, cutting-down and scaling random trees

Abstract

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 Levy tree. This extends the result obtained by Miermont [75] in the case of multitype Galton-Watson trees with finite variance.