= Peierls circuit bound for one-independent percolation
{c}
{title2=$\mathbb P(\text{specified closed dual circuit of length }\ell)\leq q^{\ell/4}$}
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 $4\ell3^\ell$ 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>.
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