Sharp dimension threshold for semicircle law in sparse high-dimensional random geometric graphs

Establish that for the high-dimensional random geometric graph G(n,d,p) formed by sampling n i.i.d. vectors uniformly on the (d−1)-dimensional unit sphere and connecting i and j when ⟨v_i,v_j⟩ ≥ τ with τ chosen so that P(⟨v_i,v_j⟩ ≥ τ)=p, the empirical spectral distribution of A/√(np(1−p)) converges to the semicircle law under the dimension condition d ≫ np log(1/p), thereby removing the extra log factor from the current proven threshold d = ω(np log^2(1/p)).

Background

The paper proves a semicircle law for the adjacency matrix of the random geometric graph G(n,d,p) under the condition d = ω(np log^2(1/p)) with np → ∞. This provides spectral indistinguishability from Erdős–Rényi graphs in sparse high-dimensional regimes.

Motivated by entropic and spectral thresholds previously identified for related problems at d ≍ np log(1/p), the authors conjecture that the extra logarithmic factor in their current condition is not necessary, suggesting that the true threshold for the semicircle law should be d ≫ np log(1/p).