Random trees, fires and non-crossing partitions

Abstract

Trees play a fundamental role in various areas of mathematics such as combinatorics, graph theory, population genetics, and theoretical computer science. In the first part of this thesis, we consider fires on large random trees. We define dynamics where, as time goes by, fires start randomly on a tree and propagate through the neighboring flammable edges, whereas in the meantime, some edges become fireproof and stop the propagation of subsequent fires. We study the effect of such dynamics, which is intimately related to the geometry of the underlying tree, and consider two different models of trees: Cayley trees and random recursive trees. The techniques used are related to the procedure of cutting-down a tree (in which edges are successively removed from the tree) and fragmentation theory. In the second part, we use trees (in particular Galton-Watson trees) as tools to study large random non-crossing configurations of the unit disk. Such configurations consist of a graph formed by non-intersecting diagonals of a regular polygon. We consider two variants: first when the connected components of the graph are pairwise disjoint polygons (the configuration is then called a non-crossing partition), and then when this graph is a tree (it is called a non-crossing tree). These objects have been studied from the perspective of combinatorics and probability but in the past, research has focused on the uniform distribution. We generalize the latter using Boltzmann sampling. In both models, we observe a universality phenomenon: all large non-crossing partitions for which the distribution of the size of a typical polygon belongs to the domain of attraction of a stable law resemble the same random object, namely a stable lamination. A similar result holds for non-crossing trees, for which we construct a novel universal limit that we call a stable triangulation.