Infinite Random Geometric Graphs from the Hexagonal Metric

We consider countably infinite random geometric graphs, whose vertices are points in R, and edges are added independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. Assume that the vertex set is randomly chosen and dense in R. We address the basic question: for what metrics is there a unique isomorphism type for graphs resulting from this random process? It was shown in [7] that a unique isomorphism type occurs for the L∞-metric for all n ≥ 1. The hexagonal metric is a convex polyhedral distance function on R2, which has the property that its unit balls tile the plane, as in the case of the L∞-metric. We may view the hexagonal metric be seen as an approximation of the Euclidean metric, and it arises in computational geometry. We show that the random process with the hexagonal metric does not lead to a unique isomorphism type.