Stationary straight-line representations of stationary random graphs

Abstract

It is a well known theorem of Thomassen that any infinite planar simple graph has a planar representation in which all edges are straight line segments that intersect only at common vertices. In this paper we put this phenomenon in a probabilistic context. An R d -representation of a graph is an embedding of the vertex set into R d . We say that a random R d -representation of a random graph is stationary if its distribution is translation-invariant, that is, the point process given by the vertex set and the edge process given by the edge relations have distributions which are invariant under translations in R d . The contribution of this paper is to give an example of a stationary R 2-representation of a random graph that possesses no stationary R 2-representation in which the edges appear as straight lines which intersect only at common vertices. Thus the natural generalization of Thomassen's theorem does not hold.