Edge-count characterization of rigidity above the critical threshold p_c

Prove that for edge probabilities p > p_c = (2 / (1 − log 2)) (log n / n), the Erdős–Rényi random graph G(n,p) is asymptotically almost surely d-rigid if and only if the event |E(G)| ≥ d n − \binom{d+1}{2} occurs, thereby characterizing rigidity solely by the edge-count condition in this regime.

Background

The paper identifies p_c as the threshold separating regimes where minimum degree versus edge count is the bottleneck for rigidity. Below p_c, the authors determine d(n,p) precisely as the minimum degree; above p_c they conjecture a sharp equivalence: d-rigidity holds exactly when the deterministic edge-count inequality is satisfied.

Their probabilistic approach (via closure size bounds) currently requires d bounded away from np/2, making it too weak to address this equivalence near the upper edge-count limit.