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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 $p_0$.

Fix $p>1/2$, and use separated-block percolation renormalization to apply the preceding one-independent bond percolation result. Place a squareat each coarse vertex $v\in {\mathbb{Z}}^2$. For a horizontal coarse edge $e=\{v,v+(1,0)\}$, let $R_e$ be the width-$5n$, height-$2n$ rectangle joining the two endpoint squares. Declare $e$ good if $R_e$ has a horizontal open crossing and both endpoint squares have vertical open crossings. For vertical coarse edges use the rotated definition.

The rectangle crossing in bond percolation limit proved above shows that each of these three events has probability tending to one. The Harris lemma givesChoose $n$ so large that every coarse bond has good probability at least the $p_0<1$ from the preceding part.

The good-bond process really is one-independent. Its determining bonds lie entirely in the corresponding $R_e$. Nonincident horizontal coarse edges either lie in rows separated by at least $3n$, while their strips have width $2n$, or in the same row separated by at least one coarse edge; their rectangles are then separated by at least $n$. The vertical case is identical. A horizontal rectangle and a vertical rectangle can intersect only if the horizontal edge's endpoint column is the vertical edge's column and the vertical edge has an endpoint in the horizontal edge's row, which would make the coarse edges incident. Thus sets of nonincident coarse edges use disjoint fine-bond sets, and those sets are independent. The positive gap avoids the shared-boundary-bond problem that occurs with touching block constructions.

An infinite good coarse cluster produces an infinite connected fine open set. To verify connectivity, a long horizontal crossing of $R_e$ contains a horizontal crossing of each endpoint square. It therefore meets every vertical crossing of that square. For collinear good edges the transverse crossing of their shared square joins their long crossings; for perpendicular good edges their long crossings meet inside the shared square. Concatenating along the coarse cluster gives a connected open fine set visiting boxes at arbitrarily large distances.

The one-independent result gives positive probability that the coarse origin belongs to an infinite good cluster. On this event some vertex of the finite square $S_0$ belongs to an infinite fine open cluster. The union bound and translation invariance of the original independent law implyso $\theta (p)>0$. Since this holds for every $p>1/2$, and the Harris theorem for square-lattice bond percolation gives $\theta (p)=0$ for $p\le 1/2$, we obtain the Harris-Kesten theorem:

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.