The critical-window conjecture for range-$R$ bond percolation

Let $V(R)$ be the number of potential range-$R$ neighbors and let $p_c(R)$ be the critical bond-percolation probability. For dimensions $d=2$ or $3$, let $\theta_c=\theta_c(d)>0$ be the critical parameter for the limiting measure-valued process described in the preceding scaling theory. The critical-window conjecture.

$$V(R)p_c(R)-1\sim\frac{\theta_c}{R^{d-1}}.$$

This conjecture refines Penrose's convergence result by identifying the first-order correction to the critical probability. It remains unresolved in the corresponding long-range contact-process setting, and the source explicitly treats this refinement as open.