arXiv subjects

Mathew D. Penrose

Publications and source records attributed to Mathew D. Penrose.

At least 19 recordsLinked to original sources

arXiv2026

Record times for coverage thresholds and maximal spacings

Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region $A \subset {\bf R}^d$ having a smooth boundary, $d \geq 1$. Let $B \subset A$ be compact. The _coverage threshold_ of $B$, $R_n$, is the smallest $r$ such that $B$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. The _maximal spacing_ $\tilde{R}_n$ is the volume of the largest ball contained in $A \setminus \{X_1,\ldots,X_n\}$. We investigate the asymptotic frequency of _record times_ in the sequence $(R_n)$, that is times $n$ for which $R_n < R_{n-1}$. Let $N_m$ denote the number of records in the sequence $(R_n)$ up to time $m$, and let $ν_m$ be the time at which the $m$th record value of the sequence $(R_n)$ occurs. For $B \subset A^o$, we show that almost surely, $N_n \sim \frac12 (\log n)^2$ and $ν_n^{1/\sqrt{n}}\to \exp \big(\sqrt{2}\: \big)$ as $n \to \infty$, and likewise for $\tilde{N}_n$ and $\tildeν_n$, defined analogously in terms of $(\tilde{R}_n)$. But if $B=A$ and $d \geq 3$, then $N_n \sim \frac12 (1- \frac{1}{d}) (\log n)^2$ and $ν_n^{1/\sqrt{n}}\to \exp \big( \sqrt{2d/(d-1)} \: \big)$. We also discuss the generalization (for fixed $k \in {\bf N}$) to $k$-coverage thresholds, maximal $k$-spacings and non-uniformly distributed points $X_i$ in $A$.

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arXiv2026

Supercritical phase of the random connection model

Given $d \in {\bf N}, λ>0$, the random connection model (RCM) in a region $A \subseteq {\bf R}^d$ is a graph with vertex set given by a homogeneous Poisson point process of intensity $λ$ in $A$, with an edge placed between each pair $x,y$ of vertices with probability $ϕ(\|x-y\|)$, where $ϕ: {\bf R}_+ \to [0,1]$ is a nonincreasing finite-range connection function. We show that if $d \geq 3$ and $λ$ is strictly supercritical for the RCM in ${\bf R}^d$, then the model remains supercritical if it is restricted to a region $A$ of the form ${\bf R}^2 \times [-K/2,K/2]^{d-2}$, provided $K$ is sufficiently large. This is a continuum analogue of a well-known result of Grimmett and Marstrand for lattice percolation. We prove this by adapting Grimmett and Marstrand's original proof; Faggionato and Hartarsky have also proved this recently by other means.

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arXiv2026

Largest component and sharpness in continuum percolation

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.

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arXiv2026

Random coverage of a manifold with boundary

Let $A$ be a compact $d$-dimensional $C^2$ Riemannian manifold with boundary, embedded in ${\bf R}^m$ where $m \geq d \geq 2$, and let $B$ be a nice subset of $A$ (possibly $B=A$). Let $X_1,X_2, \ldots $ be independent random uniform points in $A$. Define the coverage threshold $R_n$ to be the smallest $r$ such that $B$ is covered by the geodetic balls of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $A$ has Riemannian volume 1 and its boundary has surface measure $|\partial A|$, and $B=A$, then if $d=3$ then ${\bf P}[nπR_n^3 - \log n - 2 \log (\log n) \leq x]$ converges to $\exp(-2^{-4}π^{5/3} |\partial A| e^{-2 x/3})$ and $(n πR_n^3)/(\log n) \to 1$ almost surely, while if $d=2$ then ${\bf P}[n πR_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|π^{-1/2} e^{-x/2})$. We generalize to allow for multiple coverage. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform. For the limiting distribution, we have a similar result for Poisson samples. Our results still hold if we use Euclidean rather than geodetic balls.

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arXiv2026

On the components of random geometric graphs in the dense limit

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $nr_n^d$ tends to infinity but slowly enough that ${\bf E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to $μ_n$ in probability as $n \to \infty$ where (with $|A|$, $θ_d$ and $|\partial A|$ denoting the volume of $A$, of the unit $d$-ball, and the perimeter of $A$ respectively) $μ_n := ne^{-πn (r_n)^d/|A|}$ if $d=2$ and $μ_n := ne^{-θ_d n (r_n)^d/|A|} + θ_{d-1}^{-1} |\partial A| (r_n)^{1-d} e^{- θ_d n (r_n)^d/(2|A|)}$ if $d\geq 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d \geq 3$, and for Poisson input with $d \geq 2$. We extend these results (substituting ${\bf E}[S_n]$ for $μ_n$) to a class of non-uniform distributions on $A$.

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arXiv2025

Random coverage from within with variable radii, and Johnson-Mehl cover times

Given a compact planar region $A$, let $τ_A$ be the (random) time it takes for the Johnson-Mehl tessellation of $A$ to be complete, i.e. the time it takes for $A$ to be fully covered by a spatial birth-growth process in $A$ with seeds arriving as a unit-intensity Poisson point process in $A \times [0,\infty)$, where upon arrival each seed grows at unit rate in all directions. We show that if $\partial A$ is smooth or polygonal then $\Pr [ πτ_{sA}^3 - 6 \log s - 4 \log \log s \leq x]$ tends to $\exp(- (\frac{81}{4π})^{1/3} |A|e^{-x/3} -(\frac{9}{2π^2})^{1/3} |\partial A| e^{-x/6})$ in the large-$s$ limit; the second term in the exponent is due to boundary effects, the importance of which was not recognized in earlier work on this model. We present similar results in higher dimensions (where boundary effects dominate). These results are derived using new results on the asymptotic probability of covering $A$ with a high-intensity spherical Poisson Boolean model restricted to $A$ with grains having iid small random radii, which generalize recent work of the first author that dealt only with grains of deterministic radius.

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arXiv2025

On the rate of convergence in the Hall-Janson coverage theorem

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with a nice boundary. Let $α$ be the expected volume of a ball of radius $Y$, and suppose $r=r(t)$ is chosen so that $αt r^d - \log t - (d-1) \log \log t$ is a constant independent of $t$. A classical result of Hall and of Janson determines the (non-trivial) large-$t$ limit of the probability that $A$ is fully covered by $Z$. In this paper we provide an $O((\log \log t)/\log t)$ bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to $O(1/\log t)$.

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arXiv2025

On $k$-clusters of high-intensity random geometric graphs

Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime). We also show that the variance of $S_{n,k}$ is asymptotic to its mean, and prove Poisson and normal approximation results for $S_{n,k}$ in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points). We also give similar results in the so-called mildly sparse limiting regime where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $ becomes large.

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arXiv2025

Covering one point process with another

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \{Y_1,\ldots,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \to \infty$ with $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.

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arXiv2024

Fluctuations of the connectivity threshold and largest nearest-neighbour link

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ be the threshold $r$ at which the geometric graph on these $n$ vertices with distance parameter $r$ becomes $k$-connected. We show that if $d=2$ then $n (π/|A|) T_{n,1}^2 - \log n$ is asymptotically standard Gumbel. For $(d,k) \neq (2,1)$, it is $n (θ_d/|A|) T_{n,k}^d - (2-2/d) \log n - (4-2k-2/d) \log \log n$ that converges in distribution to a nondegenerate limit, where $θ_d$ is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when $(d,k)=(2,2)$ where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest $k$-nearest neighbour link $U_{n,k}$ in the sample, and show $T_{n,k}=U_{n,k}$ with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in $A$. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

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arXiv2024

On the capacity functional of the infinite cluster of a Boolean model

The original 2017 version of this paper, published in Ann. Appl. Probab., 27, 1678--1801, contains a major gap in the proofs. In the subsequent publication in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For convenience of the reader, we here update the original paper to incorporate the suggested fix. Consider a Boolean model in $R^d$ with balls of random, bounded radii with distribution $F_0$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_\infty$ is given by $θ_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact $L\subset R^d$. We prove for any fixed $L$ and $F_0$ that $θ_L(t)$ is infinitely differentiable in $t$, except at the critical value $t_c$; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution $F_0$ to vary and viewing $θ_L$ as a function of the measure $F:=tF_0$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval. We also prove that $θ_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $β$ is at most 1 (if it exists) for this model. Along the way, we extend a result of H.Tanemura (1993), on regularity of the supercritical Boolean model in $d \geq 3$ with fixed-radius balls, to the case with bounded random radii.

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arXiv2023

Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.

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arXiv2022

Giant component of the soft random geometric graph

Consider a 2-dimensional soft random geometric graph $G(λ,s,ϕ)$, obtained by placing a Poisson($λs^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices with probability $ϕ(\|x-y\|)$, where $ϕ: {\bf R}_+ \to [0,1]$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(λ,s,ϕ)$ in the large-$s$ limit with $(λ,ϕ)$ fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where $λ$ equals the critical value $λ_c(ϕ)$.

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arXiv2022

Random Euclidean coverage from within

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $A$ has volume 1 and perimeter $|\partial A|$, if $d=3$ then $\Pr[nπR_n^3 - \log n - 2 \log (\log n) \leq x]$ converges to $\exp(-2^{-4}π^{5/3} |\partial A| e^{-2 x/3})$ and $(n πR_n^3)/(\log n) \to 1$ almost surely, and if $d=2$ then $\Pr[n πR_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|π^{-1/2} e^{-x/2})$. We give similar results for general $d$, and also for the case where $A$ is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform.

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arXiv2020

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for $G(n,r_n)$ in the thermodynamic limit with $nr_n^d =$ const., and also in the dense limit with $n r_n^d \to \infty$, $r_n \to 0$. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the travelling salesman, spanning tree, matching, bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order $d-\varepsilon$, under thermodynamic scaling of the distance parameter.

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arXiv2020

Leaves on the line and in the plane

The Dead Leaves Model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For $d=2$ we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of Dead Leaves Random Measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.

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arXiv2018

Non-triviality of the vacancy phase transition for the Boolean model

In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.

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arXiv2018

Optimal Cheeger cuts and bisections of random geometric graphs

Let $d \geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on $n$ random points in a $d$-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of $n$) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large $n$ to an analogous Cheeger-type constant of the domain. Previously, García Trillos {\em et al.} had shown this for $d \geq 3$ but had required an extra condition on the distance parameter when $d=2$.

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