One-independent bond percolation

A bond percolation law is one-independent if the bond states in any two sets with disjoint endpoint sets are independent as collections. In particular, states of bonds in a matching in a graph are mutually independent. The law need not be invariant under translations or positively associated.

Place side-$2n$ square lattice boxes around the points $3nv$, $v\in {\mathbb{Z}}^2$. The width-$2n$ rectangle joining adjacent boxes has length $5n$. Declare a coarse bond open when its long rectangle is crossed and each endpoint box is crossed in the transverse direction. Nonincident coarse bonds use disjoint fine-bond sets, while incident good bonds give connected open paths inside their common endpoint box. When fixed-aspect crossing probabilities tend to one, these coarse bond probabilities tend to one. The Peierls circuit bound for one-independent percolation then yields an infinite fine-lattice open cluster.

If each bond has closed probability at most $q$ in one-independent bond percolation on the square lattice, a set of $\ell$ bonds contains a matching in a graph of size at least $\ell /4$: divide horizontal bonds by the parity of their left endpoint and vertical bonds by the parity of their lower endpoint. Each of the four classes is a matching. Mutual independence in the largest class proves the circuit bound. There are at most $4\ell 3^{\ell }$ dual circuits of length $\ell$ surrounding the origin. For sufficiently small $q$, the sum of these circuit probabilities is less than one, so the origin has a positive chance of belonging to an infinite percolation cluster.