## Electronic Journal of Probability

### Plaquettes, Spheres, and Entanglement

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1464819830

Digital Object Identifier
doi:10.1214/EJP.v15-804

Mathematical Reviews number (MathSciNet)
MR2721052

Zentralblatt MATH identifier
1229.60107

Rights

#### Citation

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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