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A path in the underlying graph all of whose edges are open. Connectivity of open paths determines the percolation clusters. A ray in a graph with all edges open is an infinite ray in a graph.

Let $E_T,E_B,E_L,E_R$ be the increasing events that an infinite simple open ray in a graph starts at the indicated side and immediately leaves the box, with all later graph vertices outside it. A corner exit is assigned according to the direction of its outgoing edge. The four events have the same probability $a_N$ by rotational symmetry. Their union is $D_N$ from the preceding step.

The Harris-FKG inequality applies to these exterior-ray events by approximation with finite connections to larger boxes and passage to decreasing limits. Positive association of the four decreasing events $E_s^c$, by iterating the Harris-FKG inequality, givesEquivalently, one may apply the preceding square-root bound for increasing events first to the two opposite-side unions and then to the two individual sides of one pair. ThereforeA union bound now showsChoose $N$ such that $1-\mathbb{P}(D_N)<(1/8{)}^4$. ThenThe infinite exterior clusters touched by these two sides need not be the same. The same argument works for dual boxes and any sufficiently large size; this permits the matching primal/dual contours used in the full proof.

Assume $\theta (1/2)>0$. The event that some infinite percolation cluster exists is invariant under lattice translations. By translation ergodicity of Bernoulli percolation, its positive probability is therefore one.

Let $D_N$ be the event that an infinite percolation cluster meets the finite graph vertex box $B_N=[-N,N{]}^2$. These increasing events increase with $N$, and their union is the event that some infinite percolation cluster exists. Continuity from below of a measure givesAn infinite percolation cluster meeting $B_N$ contains an infinite simple open ray in a graph from a graph vertex of the box. This follows from local finiteness and the König infinity lemma. That ray visits the finite box only finitely many times; its last box graph vertex lies on the boundary, and the subsequent ray is outside. Thus $D_N$ is also the event that the box boundary intersects an infinite percolation cluster, and is the union of the four exterior-side arm events.

Consequently, for every $ϵ>0$, choose $N$ withFor $ϵ\ge 1$ any sufficiently large $N$ still satisfies the strict inequality. No uniqueness assertion is needed for this preliminary step.