Electronic Journal of Probability

Plaquettes, Spheres, and Entanglement

Geoffrey Grimmett and Alexander Holroyd

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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.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 45, 1415-1428.

Accepted: 19 September 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

entanglement percolation random sphere

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Grimmett, Geoffrey; Holroyd, Alexander. Plaquettes, Spheres, and Entanglement. Electron. J. Probab. 15 (2010), paper no. 45, 1415--1428. doi:10.1214/EJP.v15-804. https://projecteuclid.org/euclid.ejp/1464819830

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