Lattice embeddings in percolation (Q662427)

Let \(D,d,M\) be positive integers. Consider the site percolation on \(\mathbb Z^d\) with parameter \(p\in [0,1],\) i.e. each site is independently open with probability \(p\). Let \(W_p(\mathbb Z^D)\) be the (random) set of all open sites, and let \(L(d,D,M,p)\) be the probability of the event that there exists an \(M\)-Lipschitz injection from \(\mathbb Z^d\) to \(W_p(\mathbb Z^D)\). Then, as main results of the paper, the critical probability \[ p_c(d,D,M):= \inf\{p:\;L(d,D,M,p)=1\}=1 \] provided either \(D\geq 2, d=2\) and \(M=1\), or \(d=D\geq 2.\) Therefore, in this case one can not ensure the existence of \(M\)-Lipschitz injection from \(\mathbb Z^d\) to \(W_p(\mathbb Z^D)\) unless for the trivial percolation with \(p=1\). Some open questions are addressed in the end of the paper for possible generalizations to the main results.