Plaquettes, Spheres, and Entanglement
Geoffrey R. Grimmett, Alexander E. Holroyd

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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Random Matrices and Applications
