Global rigidity of random graphs in ℝ

Consider the Erdős-Rényi random graph process {G_m}_m ≥ 0 in which we start with an empty graph G_0 on the vertex set [n], and in each step form G_i from G_i-1 by adding one new edge chosen uniformly at random. Resolving a conjecture by Benjamini and Tzalik, we give a simple proof that w.h.p. as soon as G_m has minimum degree 2 it is globally rigid in the following sense: For any function d E(G_m) →ℝ, there exists at most one injective function f [n] →ℝ (up to isometry) such that d(ij) = |f(i) - f(j)| for every ij ∈ E(G_m). We also resolve a related question of Girão, Illingworth, Michel, Powierski, and Scott in the sparse regime for the random graph and give some open problems.