Nicolas Broutin

The scaling limit of the minimum spanning tree of the complete graph

Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature, and contrast with the behaviour of invasion percolation on the PWIT and on regular trees, which exhibit quadratic volume growth.

Cutting down trees with a Markov chainsaw

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton-Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny

The distribution of height and diameter in random non-plane binary trees

n

The total path length of split trees

. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton-Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge.

The longest minimum-weight path in a complete graph

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees (“Otter trees”), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size n is proved to admit a limiting theta distribution, both in a central and local sense, and obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height.

Asymptotics of trees with a prescribed degree sequence and applications

We consider the model of random trees introduced by Devroye [SIAM J Comput 28, 409‐ 432, 1998]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length towards a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data structures such as binary search trees, m-ary search trees, quad trees, median-of-(2k + 1) trees, and simplex trees.

Reversing the cut tree of the Brownian continuum random tree

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8].

Partial match queries in random quadtrees

Let t be a rooted tree and nbit the number of nodes in t having i children. The degree sequence nit,ii¾?0 of t satisfies i¾?ii¾?0nit=1+i¾?ii¾?0init=|t|, where |t| denotes the number of nodes in t. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence s; we write i¾?s for the corresponding distribution. Let si¾?=nii¾?,ii¾?0 be a list of degree sequences indexed by i¾? corresponding to trees with size ni¾?i¾?+∞. We show that under some simple and natural hypotheses on si¾?,i¾?>0 the trees sampled under i¾?si¾? converge to the Brownian continuum random tree after normalisation by ni¾?1/2. Some applications concerning Galton-Watson trees and coalescence processes are provided.Copyright

Efficiently Navigating a Random Delaunay Triangulation

Consider the Aldous--Pitman fragmentation process [Ann Probab, 26(4):1703--1726, 1998] of a Brownian continuum random tree

The dual tree of a recursive triangulation of the disk

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