Topological entanglements play an important role in the physical properties, such as viscosity, of macromolecular structures. We investigate the likelihood of the appearance of entanglements in the bond percolation problem. We show that below the percolation threshold $p_c$ (but extremely close to it) there exists an entanglement threshold $p_e$. Between $p_e$ and $p_c$, there exists an infinite spanning group of interlocked (linked) clusters. Thus, in a strong gelation process, the sol-gel transition appears at $p_e$ rather than at $p_c$. We define the problem and discuss the possible approaches to its resolution. We apply the Monte Carlo renormalization-group approach to find the distance between the thresholds, and investigate some numerical characteristics of the entangled clusters. The achieved numerical resolution of $p_c$-$p_e$≊2.3×${10}^{\mathrm{-}7}$ required averaging over an extremely large numbers of configurations.

• Received 5 July 1989

DOI:https://doi.org/10.1103/PhysRevA.40.5334