Asymptotics for random Euclidean graphs: A survey (Q2752136)

This survey paper contains a very informative overview on recent results and developments in the asymptotic theory of random Euclidean graphs -- a field which becomes more and more important to various fields of applied mathematics (such as combinatorial optimization, stochastic geometry) with numerous potential applications in science and technology. A random Euclidean graph is one whose vertex set consists of i.i.d. random vectors \(X_1,\dots,X_n\) in \(R^d\), \(d\geq 2\), and one is interested in limit theorems for several functionals \(L(X_1,\dots,X_n)\) of this graph when \(n\to\infty\). Special attention has been given to (i) strong laws of large numbers for the total edge length, (ii) rates of convergence of mean values \(EL(X_1,\dots,X_n)\), (iii) central limit theorems and large deviation principles. Moreover, the author gives a CLT for the total edge length in Poisson-Voronoi tessellations and sketches the applicability of similar CLTs to other graphs in computational geometry. The paper ends with a collection of prominent open problems in this field.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00043].