Topological entanglements in the percolation problem

Abstract

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 ${\mathrm{p}}_{\mathrm{c}}$ (but extremely close to it) there exists an entanglement threshold ${\mathrm{p}}_{\mathrm{e}}$. Between ${\mathrm{p}}_{\mathrm{e}}$ and ${\mathrm{p}}_{\mathrm{c}}$, there exists an infinite spanning group of interlocked (linked) clusters. The achieved numerical resolution of ${\mathrm{p}}_{\mathrm{c}}$-${\mathrm{p}}_{\mathrm{e}}$=(1.8\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}7}$ required averaging over an extremely large number of configurations.