Abstract
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 $p_e\geq \mu^{-2}$ where $\mu$ is the connective constant for self-avoiding walks on $\mathbb{Z}^3$. Furthermore, when the edge density $p$ is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.
Citation
Geoffrey Grimmett. Alexander Holroyd. "Plaquettes, Spheres, and Entanglement." Electron. J. Probab. 15 1415 - 1428, 2010. https://doi.org/10.1214/EJP.v15-804
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