Plaquettes, Spheres, and Entanglement

The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d − 1)- sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d = 3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e ≥ µ⁻² where µ is the connective constant for self-avoiding walks on Z³ . Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.