On the Number of Entangled Clusters

We prove that the number of entangled clusters with N edges in the simple cubic lattice grows exponentially in N. This answers an open question posed by Grimmett and Holroyd (Proc. Lond. Math. Soc. 81:485–512, 2000). Our result has immediate implications for entanglement percolation: we obtain an improved rigorous lower bound on the critical probability, and we prove that the radius of the entangled component of the origin has exponentially decaying tail when p is small.