Entanglement percolation and spheres in ${\mathbb{Z}}^d$

Abstract

We obtain a new lower bound of 0.06576 for the 1-entanglement critical probability (in dimension 3), and prove that the critical point for the existence of a sphere surrounding the origin and intersecting only closed bonds in ${\mathbb{Z}}^d$ is greater than $\frac{1}{8(d-1)}$, $d\ge 3$. This substantially improves the previous lower bounds and gives the correct order of magnitude for large $d$.