See Percolation theory

Continuity of the order parameter $P_\infty (p)$ (the probability that an occupied site is in the infinite cluster for a given occupancy $p$) at $p_c$ is an open mathematical problem in the general case, but it is known to hold rigorously in 2D and $d \geq 19$ using lace expansion methods (Mean-Field Behaviour and the Lace Expansion). The conjecture that $P_\infty (p) ( p = p_c ) = 0$ for $3 \leq d \leq 18$ remains however one of the open problems in the field (V. Beffara, V. Sidoravicius, Percolation Theory).