Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 2 / b

The number $N_{\infty }$ is invariant under lattice automorphisms, so part a makes it almost surely equal to one constant in $\{0,1,2,\dots ,\infty \}$. For $p>p_c$ the constant is nonzero.

Suppose it were a finite $k\ge 2$. As boxes increase to ${\mathbb{Z}}^d$, with positive probability one box meets all $k$ infinite clusters. On that event, open a finite collection of edges inside the box joining those clusters. The finite-energy property of Bernoulli percolation gives the modified event positive probability, but it has fewer than $k$ infinite clusters. This contradicts almost-sure constancy. Hence the only possibilities are