We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and, in particular, in communications.
Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.
We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, an…
We define and analyze an extension to the $d$-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic directi…
We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius $r$ grows sublinearly with $r$. Then, we prove that in each (deterministic) direction, there exists with probability one a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there are more than one semi-infinite paths is dense in $[0,2π)$. It corresponds to possi…
The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process $\mathcal{N}$ on $\mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(u)$ of a vertex $u \in \mathcal{N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjectu…
We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to infinity is the Poisson(1) branching process conditioned to survive forever. An extension to "almost" regular graphs and a quenched version are also given.
Since the works of Howard and Newman (2001), it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path. Moreover, there exists a set of 'exceptionnal' directions reached by (at least) two infinite paths which is random, dense and only countable in dimension 2. Howard and Newman's method says nothing about (random) directions reached by more than two infinite paths and, in particular, if such 'very exceptionnal' directions exist in dimension 2. In this paper, we pr…
This note answers one question in [math.PR/0505668], concerning the connected allocation for the Poisson process in R^2. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.
For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high probability (log n/n)^(1/d).
We investigate asymptotics for the minimal spanning acycles of the (Alpha)-Delaunay complex on a stationary Poisson process on $\mathbb{R}^d, d \geq 2$. Minimal spanning acycles are topological (or higher-dimensional) generalization of minimal spanning trees. We establish a central limit theorem for total weight of the minimal spanning acycle on a Poisson-Delaunay complex. Our approach also allows us to establish central limit theorems for sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of m…
We introduce a system of coalescing random paths with radialbehavior in a subsetof the plane. We call it theDiscrete Radial Poissonian Web. We show that underdiffusive scaling this family converges in distribution toa mapping of a restrictionof the Brownian Web.
Kesten and Lee [36] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^d$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative? However, techniques employed to tackle the same problem for other functionals studied in geometric probability do not apply directly to the minimal spanning tree. Thus the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees has remained open. In this work, we establish bounds on the convergence rate fo…
We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these r…
We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic Poisson-Voronoi mosaic, $Y$ has a radial and log-concave distribution, implying that ${|Y|}/{\mathbb{E}(|Y|^2)^{\frac{1}{2}}}$ approaches one for large $n$. Assuming the cell intensity of the random mosaic scales like $e^{n ρ_n}$, where $\lim_{n \to \infty} ρ_n = ρ$, $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that…
We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one - that we evaluate and compare at two different scales - and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization - see Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof …
We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state-space of abstract graphs with real-valued edge-weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in ord…
We study the total $α$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in Bhatt and Roy (2004) - on a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d \ge 3$. While a Dickman limit was proved in Penrose and Wade (2004) in the case of $d=2$, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when $α=1$, with a rate of convergence of the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we extend these results and prove a central limit theorem in any dimen…
The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\PGW(λ)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $λ$. Suppose that $λ^*>λ>1$. We show that $\PGW(λ^*)$ conditioned to survive forever …
Is there a constant $r_0$ such that, in any invariant tree network linking rate-$1$ Poisson points in the plane, the mean within-network distance between points at Euclidean distance $r$ is infinite for $r > r_0$? We prove a slightly weaker result. This is a continuum analog of a result of Benjamini et al (2001) on invariant spanning trees of the integer lattice.
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 flu…
A weighted $d-$complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and \emph{death times} in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted $d-$compl…
Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from \textit{directed spanning forests} (DSF) based on randomly perturbed lattices, which have desirable statistical properties as…
The metric $D_α(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^α)^{1/α}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics when $α> 1$. The cases $1 <α< \infty$ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors while the geodesics in the case $α= \infty$ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conj…
We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D…
We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov $\rm{HL}(0)$ model. We describe the asymptotic…
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $δ\to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $δ\Z^2$, the minimal spanning tree on $δ\Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $δ^{-2}$. In each case, sample trees are p…
Uniform spanning trees and random matrix statistics
University of Vienna（维也纳大学）；Vienna University of Technology（维也纳工业大学）；Capital Normal University, Beijing（首都师范大学）；Aalto University and University of Bonn（阿尔托大学和波恩大学）
