Existence of a phase transition for entanglement percolation

We consider the bond percolation model on the three-dimensional cubic lattice, in which individual edges are retained independently with probability p. We shall describe a subgraph of the lattice as 'entangled' if, roughly speaking, it cannot be 'pulled apart' in three dimensions. We shall discuss possible ways of turning this into a rigorous definition of entanglement. For a broad class of such definitions, we shall prove that for p sufficiently close to zero, the graph of retained edges has no infinite entangled subgraph almost surely, thereby establishing that there is a phase transition for entanglement at some value of p strictly between zero and unity.