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Entanglement percolation and spheres in $\mathbb{Z}^d$
2021-12-07 · via math.PR updates on arXiv.org

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 $\frac1{8(d-1)}$, $d \geq 3$. This substantially improves the previous lower bounds and gives the correct order of magnitude for large $d$.

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