# Plaquettes, Spheres, and Entanglement - Mathematics > Probability

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Abstract: The high-density plaquette percolation model in d dimensions contains asurface that is homeomorphic to the d-1-sphere and encloses the origin. Thisis proved by a path-counting argument in a dual model. When d=3, this permitsan improved lower bound on the critical point p e of entanglement percolation,namely p e >= \mu^-2 where \mu is the connective constant for self-avoidingwalks on Z^3. Furthermore, when the edge density p is below this bound, theradius of the entanglement cluster containing the origin has an exponentiallydecaying tail.

Author: ** Geoffrey R. Grimmett, Alexander E. Holroyd**

Source: https://arxiv.org/