Entanglement in percolation (Q2766373)

Let \({\mathbb Z}^3\) be the set of all 3-vectors \(x=(x_1,x_2,x_3)\) of integers and \({\mathbb L}=\{\{x,y\}\subset {\mathbb Z}^3: \|x-y\|=1\},\) where \(\|\cdot\|\) denotes Euclidean distance. Some definitions for finite and infinite entangled graphs in \({\mathbb L}\) are considered. The existence (or not) of such objects in bond percolation (with density \(p\)) on \({\mathbb L}\) is studied. The main results of the paper are: (i) The entanglement critical probabilities are defined. It is proven that, for \(p\) sufficiently small, the tail of the distribution of the radius of the finite entangled graph at the origin decays ``near-exponentially''. (ii) It is proven that, if \(p\) is sufficiently close to 1, there exists almost surely a unique infinite maximal entangled graph. Also it is shown the uniqueness of the infinite maximal entangled graph when \(p\) is greater than the connectivity critical probability \(p_c\), for a particular definition of infinite entanglement.