Stationary straight-line representations of stationary random graphs

AbstractIt 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 Rd-representation of a graph is an embedding of the vertex set into Rd. We say that a random Rd-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 Rd. The contribution of this paper is to give an example of a stationary R2-representation of a random graph that possesses no stationary R2-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