Plaquettes, Spheres, and Entanglement

TL;DR

This paper investigates the topological properties of high-density plaquette percolation in multiple dimensions, providing bounds on entanglement percolation thresholds and analyzing cluster size decay.

Contribution

It introduces a path-counting argument to prove the existence of spherical surfaces in the model and improves bounds on the entanglement critical point in three dimensions.

Findings

Existence of spherical surfaces in high-density plaquette percolation

Lower bound on entanglement critical point p_e >= μ^-2 in 3D

Exponential decay of cluster radius below the critical point

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 >= \mu^-2 where \mu is 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.