Rigorous explanation for aspect-ratio effects on Hopf vs. Borromean percolation thresholds

Provide a rigorous explanation for why, in dense random rectangle networks with fixed rectangle area, the Hopf-link percolation threshold appears approximately independent of rectangle aspect ratio while the Borromean percolation threshold decreases with aspect ratio.

Background

Empirical results indicate that Hopf-link percolation thresholds are largely insensitive to rectangle aspect ratio, whereas Borromean thresholds decrease as rectangles become more elongated. The authors offer heuristic arguments but no formal derivation.

Understanding the mechanistic basis of these contrasting dependencies would clarify the geometry–topology interplay governing two-body (Hopf) versus three-body (Borromean) connectivity in dense, disordered networks.