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

TL;DR

This paper establishes new lower bounds for entanglement percolation thresholds and sphere existence in high-dimensional integer lattices, advancing understanding of phase transitions in these models.

Contribution

It provides improved lower bounds for the 1-entanglement critical probability in 3D and the sphere crossing threshold in $ olinebreak bZ^d$, refining previous estimates.

Findings

Lower bound of 0.06576 for 1-entanglement critical probability in 3D

Critical point for spheres exceeds 1/(8(d-1)) for d ≥ 3

Bounds are asymptotically tight for large dimensions

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