Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 13 / 6 / iii / Solution

Put $q=1-p_0$. In one-independent bond percolation, bond states in two sets with disjoint endpoint sets are independent as collections. In particular, the states of any finite matching in a graph are mutually independent, by induction on its size. Neither translation invariance nor the Harris lemma is assumed for this law.

If the origin's open cluster is finite, the outer boundary of the union of its vertex-centered unit squares contains a closed dual circuit surrounding the origin. A specified circuit of length $\ell$ corresponds to $\ell$ distinct closed primal bonds. Partition those bonds into four matchings in a graph: horizontal bonds according to the parity of the left endpoint's horizontal coordinate, and vertical bonds according to the parity of the lower endpoint's vertical coordinate. Some class has at least $\ell /4$ bonds. Their mutual independence and their individual closed probabilities at most $q$ implyThis is the Peierls circuit bound for one-independent percolation.

There are at most $4\ell 3^{\ell }$ dual circuits of length $\ell$ enclosing the origin. To obtain this crude bound, choose a crossing of the positive horizontal ray; its distance from the origin is at most $\ell$, since the circuit also extends to the other side of the origin. There are at most $\ell$ choices for its dual edge, at most two orientations, and at most three choices at each subsequent step of a nonbacktracking walk. The stated bound allows extra slack and overcounting. The union bound now givesFor instance choose $p_0=1-10^{-8}<1$, so $x=3q^{1/4}=0.03$. The exact sum isIt follows that $\tilde{\mathbb{P}}(|C_0|=\infty )>0$. This deliberately nonoptimal constant suffices for existence of a universal $p_0$.