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Nicolas Broutin

Publications and source records attributed to Nicolas Broutin.

At least 37 records · Page 2Linked to original sources
arXiv2015

The dual tree of a recursive triangulation of the disk

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224-2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov-Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall.

math.PR↗
arXiv2014

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

We 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 $λ$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $(B_t+λt - t^2/2, t\geq 0)$, 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.

math.PR↗
arXiv2014

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 $n$. 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.

math.PR↗
arXiv2014

Cutting down $\mathbf p$-trees and inhomogeneous continuum random trees

We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the ICRTs (scaling limits of $\mathbf p$-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation. The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree.

math.PR↗
arXiv2014

Almost optimal sparsification of random geometric graphs

A random geometric irrigation graph $Γ_n(r_n,ξ)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_1,\ldots,X_n$ in the unit square $[0,1]^2$. Each point $X_i$ selects $ξ_i$ neighbors at random, without replacement, among those points $X_j$ ($j\neq i$) for which $\|X_i-X_j\| < r_n$, and the selected vertices are connected to $X_i$ by an edge. The number $ξ_i$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_i$ such that $ξ_i$ satisfies $1\le ξ_i \le κ$ for a constant $κ>1$. We prove that when $r_n = γ_n \sqrt{\log n/n}$ for $γ_n \to \infty$ with $γ_n =o(n^{1/6}/\log^{5/6}n)$, then the random geometric irrigation graph experiences explosive percolation in the sense that when $\mathbf E ξ_i=1$, then the largest connected component has size $o(n)$ but if $\mathbf E ξ_i >1$, then the size of the largest connected component is with high probability $n-o(n)$. This offers a natural non-centralized sparsification of a random geometric graph that is mostly connected.

math.PR↗
arXiv2014

Efficiently navigating a random Delaunay triangulation

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy cone walk. We prove that given $n$ uniform points in a smooth convex domain of unit area, and for any start point $z$ and query point $q$; cone walk applied to $z$ and $q$ will access at most $O(|zq|\sqrt{n} +\log^7 n)$ sites with complexity $O(|zq|\sqrt{n} \log \log n + \log^7 n)$ with probability tending to 1 as $n$ goes to infinity. We additionally show that in this model, cone walk is $(\log ^{3+ξ} n)$-memoryless with high probability for any pair of start and query point in the domain, for any positive $ξ$. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.

math.PR↗
arXiv2014

Connectivity of sparse Bluetooth networks

Consider a random geometric graph defined on $n$ vertices uniformly distributed in the $d$-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" $r_n$. We consider {\sl Bluetooth networks} that are locally sparsified random geometric graphs. Each vertex selects $c$ of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of $n^{-(1-δ)/d}$ for some $δ> 0$, then a constant value of $c$ is sufficient for the graph to be connected, with high probability. It suffices to take $c \ge \sqrt{(1+ε)/δ} + K$ for any positive $ε$ where $K$ is a constant depending on $d$ only. On the other hand, with $c\le \sqrt{(1-ε)/δ}$, the graph is disconnected, with high probability.

math.PR↗
arXiv2013

A limit process for partial match queries in random quadtrees and $2$-d trees

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(ξ)$ to visit in order to report the items matching a random query $ξ$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf {E}[{C_n(ξ)}]\simκn^β$, where $κ$ and $β$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit us to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf {E}[{\max_{x\in[0,1]}C_n(x)}]\sim γn^β$; this settles a question of Devroye [Pers. Comm., 2000].

math.PR↗
arXiv2013

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

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erdös-Rényi random graph. We exploit the explicit description of the scaling limit of the Erdös-Rényi random graph in the so-called critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST.

math.PR↗
arXiv2012

The total path length of split trees

We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) 409-432]. 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 toward 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.

math.PR↗
arXiv2012

Asymptotics of trees with a prescribed degree sequence and applications

Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The degree sequence $(n_i(t),i\geq 0)$ of $t$ satisfies $\sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, where $|t|$ denotes the number of nodes in $t$. In this paper, we consider trees sampled uniformly among all trees having the same degree sequence $\ds$; we write $`P_\ds$ for the corresponding distribution. Let $\ds(κ)=(n_i(κ),i\geq 0)$ be a list of degree sequences indexed by $κ$ corresponding to trees with size $\nk\to+\infty$. We show that under some simple and natural hypotheses on $(\ds(κ),κ>0)$ the trees sampled under $`P_{\ds(κ)}$ converge to the Brownian continuum random tree after normalisation by $\nk^{1/2}$. Some applications concerning Galton--Watson trees and coalescence processes are provided.

math.PR↗
arXiv2012

Longest path distance in random circuits

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result.

math.PR↗
arXiv2011

Partial match queries in random quadtrees

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quad trees and k-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(ξ)$ to visit in order to report the items matching an independent and uniformly on $[0,1]$ random query $ξ$ satisfies $\Ec{C_n(ξ)}\sim κn^β$, where $κ$ and $β$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(x)$ of any fixed query $x\in [0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $E{\max_{x\in [0,1]} C_n(x)} \sim γn^β$.

math.PR↗
arXiv2011

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

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, as well as 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.

math.PR↗
arXiv2011

Connectivity threshold for Bluetooth graphs

We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius $r$ that determines the "visible neighbors" of each node and the number of edges $c$ that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters $r, c$ and completely characterize the connectivity threshold (in $c$) for values of $r$ close the critical value for connectivity in the underlying random geometric graph.

math.PR↗
arXiv2010

On combinatorial testing problems

We study a class of hypothesis testing problems in which, upon observing the realization of an $n$-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been ``contaminated,'' that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples.

math.ST↗
arXiv2009

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is negative, together with its entire descendence. The size $Z$ of the set of un-killed particles is almost surely finite. In this paper, we confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log Z]=infinity. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.

math.PR↗