Infinite Random Geometric Graphs

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set in R equipped with the metric derived from the L∞-norm, then it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GRn, is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GRn. In contrast, we show that infinite random geometric graphs in R with the Euclidean metric are not necessarily isomorphic.