“…The current best bound, due to Brennan, Bresler, and Nagaraj [BBN20], asserts that in the regime p = Θ 1 n , d TV (Geo d (n, p), G(n, p)) → 0 so long as d ≫ n 3/2 . In essence, their bound relies on the fact that independent random vectors v i , v j ∼ S d−1 have |〈v i , v j 〉| = O( 1 d ) with high probability; when d ≫ n 3/2 , these inner products are small enough relative to n that v i has negligible projection (of order ≈ j =i 〈v i , v j 〉 2 = O( n/d)) into span{v j } j =i , which is enough to guarantee approximate independence of edges.…”