Crossing sharpness and subcritical decay for square-lattice FK

By [A1] and [F1] the free and wired probability laws exist. In either law the events $D_r$ decrease with $r$. Their intersection is precisely the event that the origin cluster is unbounded: connection to every box boundary is equivalent to an unbounded cluster, and every finite path reaching a farther boundary contains an initial path to the nearer boundary. Countable additivity gives continuity from above, hence ${\theta }^b(p,q)={\inf }_r{\pi }_p^b(D_r)={\mathrm{\Theta }}^b(p)$. This uses actual existing probability laws, rather than inferring their existence from consistent numerical cylinder weights.

Fix $p>p_{\mathrm{s}\mathrm{d}}(q)$ and set $t=(p+p_{\mathrm{s}\mathrm{d}}(q))\mathrm{/}2$. Then $p_{\mathrm{s}\mathrm{d}}(q)<t<p$. By [F2] and [F3], ${\mathrm{\Theta }}^0(p)\ge {\mathrm{\Theta }}^1(t)>0$, while [F3] directly gives ${\mathrm{\Theta }}^1(p)>0$. Step 1.1 proves positive percolation for both laws at $p$. No equality of free and wired laws is used.

For $p<p_{\mathrm{s}\mathrm{d}}(q)$ apply [F4]. In particular, each free or wired law satisfies the displayed bound. An origin path to $\mathrm{\partial }{\mathrm{\Lambda }}_r$ reaches one of its $8r$ vertices, each of Euclidean norm at least $r$. A finite union bound gives $\varphi (D_r)\le 8rC(p,q)e^{-c(p,q)r}\to 0$. Step 1.1 then gives zero percolation. Both defining infima are $p_{\mathrm{s}\mathrm{d}}(q)$, since every parameter above it belongs to each positive-percolation set and none below it does. The bound for all other cylinder-limit states is already exactly [F4].