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D. J. A. Welsh coauthored a study of square-lattice bond percolation with P. D. Seymour. Their crossing-probability arguments relate crossings of squares to crossings of rectangles. These estimates help control connectivity across different scales and form part of Russo-Seymour-Welsh theorem.^[ref]

In independent bond percolation on the square lattice at $p=1/2$, write $\pi (r)$ for the one-arm probability to maximum-norm distance $r$. There is a constant $c>0$, independent of $r$, such thatTo see the geometry, require an open path to distance $r$, an open graph cycle surrounding the origin in an annulus between distances about $r/2$ and $3r/4$, and a long crossing from inside that cycle to distance $2r$. The initial arm and the long crossing both intersect the cycle, producing a connected route to the larger distance.

The Russo-Seymour-Welsh theorem bounds the cycle and long-crossing probabilities below by positive constants because their aspect ratios remain fixed. All the required events are increasing, so the Harris-FKG inequality bounds their joint probability below by a constant times $\pi (r)$. Integer rounding affects only the constants. The argument extends an existing arm with uniformly positive conditional cost; it does not bound the arm event itself below by a positive constant.^[ref]

P. D. Seymour studied square-lattice bond percolation with D. J. A. Welsh. Their work constructs crossing bounds for rectangles from square-crossing information, using positive correlation and geometric combinations of crossing events. Such scale comparisons are central to Russo-Seymour-Welsh theorem and planar critical connectivity.^[ref]

For critical independent bond percolation on the square lattice, two one-arm events can be joined at a scale-independent additional cost. With $y=(r,0)$, there is $c>0$ such thatThe one-arm probability factors remain present: this is a lower bound proportional to the two arm probabilities, not a uniform positive bound for the unconditional connection probability.

For a concrete construction at large integer $r$, put $s=\lfloor r/12\rfloor$. Require arms from $0$ and $y$ to distance $4s$, open graph cycles around each root between distances $2s$ and $4s$, and a left-to-right crossing of $[-s,r+s]\times [-s,s]$. Each end of that crossing lies inside the corresponding cycle, so it crosses both cycles. Each arm also meets its cycle, and the combined configuration connects the roots.

The Russo-Seymour-Welsh theorem gives fixed positive lower bounds for the cycles and corridor crossing. The Harris-FKG inequality gives a joint lower bound $c\pi (4s{)}^2\ge c\pi (r{)}^2$. Finitely many small $r$ can be absorbed into the constant. This makes the geometric and probability costs of the gluing explicit.^[ref]

For independent bond percolation on the square lattice at $p=1/2$, the Russo-Seymour-Welsh theorem gives nontrivial crossings at every scale. For each fixed aspect ratio $\rho \ge 1$, there is $c_{\rho }>0$ such thatwhere $h_p(m,n)$ is the left-to-right rectangle crossing in bond percolation probability. The constant can depend on $\rho$, but not on the rectangle's size.

Reflection extensions and transverse crossings in overlapping rectangles permit longer crossings to be assembled from square crossings. The Harris-FKG inequality controls intersections of these increasing events; dual bond percolation supplies the complementary crossing bound. These are the geometric ingredients of the original square-lattice crossing analysis.^[ref]

For example, a rectangle twice as wide as it is high retains a positive crossing probability as both dimensions grow. The bounds imply positive probabilities for open or closed circuits in fixed-shape annuli and support the one-arm extension estimate. Their critical-parameter assumption is essential: a corresponding uniform lower bound would contradict exponential decay of subcritical percolation.