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Uniform fixed-aspect rectangle crossing in bond percolation bounds at the self-dual parameter yield a positive lower bound for closed dual circuits in geometrically spaced disjoint annuli. The independent annular barriers for percolation imply that an origin-to-infinity open graph path has probability zero. Parameter monotonicity gives zero percolation probability for every $p\le 1/2$.

We first derive the reflected attachment estimate for percolation. Put $R=[0,m]\times [0,2n]$, $m\ge n$, and $S=[0,n{]}^2$. Let $X(R)$ require an open vertical crossing of $S$ together with an open path in $R$ attaching it to the right side of $R$. ThenTo prove this, explore the leftmost vertical open crossing $P$ of $S$, revealing bonds on $P$ and to its left, but none strictly to its right. One can implement this exploration by following the boundary of the set accessible from the left using closed dual bonds; its right boundary is the extremal open crossing when that crossing exists. Thus fixing its outcome leaves right-side bonds in their original independent distribution.

For each possible explored $P$, reflect its geometry in $y=n$ and join it to its reflection to obtain a bottom-to-top separator $L$ of $R$. This reflected portion is only a geometrical separator and is not presumed open. Let $Y_{-}$ and $Y_{+}$ mean that an open path from the right boundary reaches, respectively, the lower or upper half of $L$, staying to its right until its endpoint. Any horizontal open crossing of $R$, followed from right to left until its first meeting with $L$, witnesses at least one of these events. Reflection in $y=n$ swaps $Y_{-}$ and $Y_{+}$ and preserves the independent measure. HenceThe bonds tested by $Y_{-}$ remain unexposed after specifying $P$; conditional on that exploration its probability is still the unconditional one for this deterministic separator. Its occurrence attaches the genuinely open lower crossing $P$ to the right side. Average over $P$ to obtain the attachment estimate.

Now take $R=[0,2n]\times [0,2n]$ and $R^{\prime }=[-n,n]\times [0,2n]$, with common corner square $S=[0,n{]}^2$. Use $X(R)$, its horizontal reflection in $R^{\prime }$, and a horizontal open crossing of $S$. The last crossing meets the vertical crossings of $S$ in both attachment events, so these three events give a horizontal crossing of $R\cup R^{\prime }$. All three events are increasing events, so the Harris lemma gives

For all aspect ratios, glue a width-$m_1$, height-$2s$ rectangle to a width-$m_2$, height-$2s$ rectangle with an overlap square of side $2s$. Their horizontal crossings and an overlap vertical crossing imply a crossing of their union. Again the Harris lemma givesAdding a $3s$-wide rectangle therefore yieldsFor height $n\ge 2$, let $s=\lfloor n/2\rfloor$, so $2s\le n$ and $s\ge n/3$. Choose an integer $k\ge \max (3,3\lambda )$. For $m\le \lambda n$, we have $m\le ks$; restricting to height $2s$ and extracting a shorter horizontal crossing givesFor $n=1$, the horizontal bottom-row path gives $h_{1/2}(m,1)\ge 2^{-\lceil \lambda \rceil }$ whenever the allowed integer $m\ge 1$ exists. Thus one may take $c_{\lambda }=\min (2^{17-8k},2^{-\lceil \lambda \rceil })>0$. This proves the requested uniform rectangle crossing in bond percolation bound, rather than assuming the Russo-Seymour-Welsh theorem.

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: