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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 μis the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.