The maximal degree in a Poisson–Delaunay graph

Abstract

We investigate the maximal degree in a Poisson-Delaunay graph in R d , d ≥ 2, over all nodes in the window Wρ := ρ 1/d [0, 1] d as ρ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting d = 2, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when d ≥ 3.We summarize here the notation used throughout the text. General notationWe denote by N = {1, 2, . . .} and R + = [0, ∞) the sets of positive integers and nonnegative numbers, respectively. The d-dimensional Euclidean space R d is endowed with the Euclidean norm · and with its d-dimensional Lebesgue measure V d (·). We denote by B d + the set of Borel sets B ⊂ R d such that 0 < V d (B) < ∞. The unit sphere with dimension d − 1 is denoted by S d−1 .

Limit laws for large th-nearest neighbor balls

“…When dealing with limit laws for large kth-nearest neighbor distances of a sequence of i.i.d. random points in R d with density f , which take values in a bounded region K, the modification of the kth-nearest neighbor distances made in (6) (by introducing the 'boundary distances' X i − ∂K ) and the condition that f is bounded away from zero, which have been adopted in [12] and [13], seem to be crucial, since boundary effects play a decisive role [8,9]. Regarding kth-nearest neighbor balls with large probability volume, there is no need to introduce X i − ∂K .…”

“…be the event that each of the subcubes contains at least one of the points of X n . The event E n is extensively used in stochastic geometry to derive central limit theorems or to deal with extremes [3,6,7], and it will play a crucial role throughout the rest of the paper. The following lemma, which captures the idea of 'asymptotic independence', is at the heart of our development.…”

Large degrees in scale-free inhomogeneous random graphs

“…The behaviour of the maximum degree for the random graphs considered in this paper is significantly different from that of the maximum degree of an Erdős-Rényi graph or a random geometric graph, which is concentrated with a high probability on at most two consecutive numbers (see [3,Theorem 3.7] and [17,Theorem 6.6] or [15], respectively). More recently, such a concentration phenomenon was also shown for the maximum degree of a Poisson-Delaunay graph in [4].…”