Martingales and Profile of Binary Search Trees

We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile

Models of random subtrees of a graph

Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. A subtree of $G$ with size $n$ is a tree which is a subgraph of $G$, with $n$ vertices. When $n=N$, such a subtree is called a spanning tree. The main purpose of this paper is to explore the question of uniform sampling of a subtree of $G$, or a subtree of $G$ with a fixed number of nodes $n$, for some $n\leq N$. We provide asymptotically exact simulation methods using Markov chains. We highlight the case of the uniform subtree of $\Z^2$ with $n$ nodes, containing the origin $(0,0)$ for which Schramm asked several questions. We produce pictures, statistics, and some conjectures.\par The second aim of the paper is devoted to survey other models of random subtrees of a graph, among them, we will discuss DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide a number of new models, some statistics, and some conjectures

Parametrised branching processes: a functional version of Kesten & Stigum theorem

Let (Z_n , n ≥ 0) be a supercritical Galton-Watson process whose offspring distribution µ has mean λ > 1 and is such that int_{x>0} x log(x) dµ(x) 1 for all λ ∈ I. This allows us to define Z_n (λ) the number of elements in the nth generation at time λ. Set W_n (λ) = Z_n (λ)/λ^n for all n ≥ 0 and λ ∈ I. We prove that, under some moment conditions on the process X, the sequence of processes (W_n (λ), λ ∈ I) n≥0 converges in probability as n tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation

A new encoding of coalescent processes. Applications to the additive and multiplicative cases

International audienceWe revisit the discrete additive and multiplicative coalescents, starting with n particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to\infty$. We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous [Ann. Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $\lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $B_t+\lambda t−t^2/2,t≥0)$, but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process

Aldous-Broder theorem: extension to the non reversible case and new combinatorial proof

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph G, but it is more general: it states that given a reversible M Markov chain on G started at r, the tree rooted at r formed by the steps of successive first entrance in each node (different from the root) has a probability proportional to $\prod_{e=(e1,e2)\in{\sf Edges}(t,r)} M_{e1,e2}$ , where the edges are directed toward r. As stated, it allows to sample many distributions on the set of spanning trees. In this paper we extend Aldous-Broder theorem by dropping the reversibility condition on M. We highlight that the general statement we prove is not the same as the original one (but it coincides in the reversible case with that of Aldous and Broder). We prove this extension in two ways: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences