Local and Global Expansion in Random Geometric Graphs

Consider a random geometric 2-dimensional simplicial complex X sampled as follows: first, sample n vectors u1,…,un uniformly at random on Sd−1; then, for each triple i,j,k ∈ [n], add i,j,k and all of its subsets to X if and only if ⟨ ui,uj ⟩ ≥ τ, ⟨ ui,uk ⟩ ≥ τ, and ⟨ uj, uk ⟩ ≥ τ. We prove that for every ε > 0, there exists a choice of d = Θ(logn) and τ = τ(ε,d) so that with high probability, X is a high-dimensional expander of average degree nε in which each 1-link has spectral gap bounded away from 1/2.