Fires on large recursive trees

We consider random dynamics on a uniform random recursive tree with $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as $n\to\infty$, and prove a phase transition when $p_n$ is of order $\ln n/n$. We then study the connectivity of the fireproof forest, more precisely the existence of a giant component. We finally investigate the sizes of the burnt subtrees.Comment: Accepted for publication in Stochastic Processes and their Applications. 24 pages, 4 figure

Triangulating stable laminations

We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.Comment: 34 pages, 5 figure

Simply generated non-crossing partitions

Differs slightly from the published version.International audienceWe introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with constraints on their block sizes. Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of size $k$ are in correspondence with vertices of out-degree $k$. This allows us to obtain limit theorems concerning the block structure of simply generated non-crossing partitions. We apply our results in free probability by giving a simple formula relating the maximum of the support of a compactly supported probability measure on the real line in terms of its free cumulants

On scaling limits of planar maps with stable face-degrees

We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$. We prove that when conditioning such maps to have $n$ vertices, or $n$ edges, or $n$ faces, the vertex-set endowed with the graph distance suitably rescaled converges in distribution towards the celebrated Brownian map when $\alpha=2$, and, after extraction of a subsequence, towards another `$\alpha$-stable map' when $\alpha <2$, which improves on a first result due to Le Gall & Miermont who assumed slightly more regularity.Comment: 31 pages, 5 figure

Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps

We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaym\'e-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map.Comment: Compared to V2 we only changed the presentation: several theorems have been merged and are now stated in a unified way; also the previous section on maps has been split into a section on trees and another one on maps only; last the former technical section 4 has moved to Appendix

Random trees, fires and non-crossing partitions

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 dfine dynamics where, as time goes by, fires start randomly on a tree and propagate through the neighboring fiammable edges, whereas in the meantime, some edges become fireproof and stop the propagation of subsequent fires. We study the efiect of such dynamics, which is intimately related to the geometry of the underlying tree, and consider two difierent 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 nonintersecting 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 noncrossing 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