Periodic-torus RSW lower bounds at the FK self-dual point

The invertible linear map $u=x+y$, $v=y-x$ sends the ordinary square lattice to $P$ and its nearest-neighbor edges to the diagonal edges. Its inverse is $x=(u-v)\mathrm{/}2$, $y=(u+v)\mathrm{/}2$. Ordinary periods $(m,0),(0,m)$ become $(m,-m),(m,m)$, generating exactly the diagonal-period lattice in [F4]. It is therefore a labelled graph isomorphism, preserving the weights in [F1]. A rotated patch with both coordinate spans less than $m$ embeds: every nonzero period has at least one coordinate of absolute value at least $m$. We may compute probabilities of such patches in these coordinates. The cellular torus comparison [F3] is the likelihood input used in [F4], not an unproved planar substitution for torus topology.

A closed rotated vertical rectangle of width $S\in 4{\mathbb{Z}}_{>0}$ and height $2S$ has crossing probability at least $d=b^{2}a$ if $2S<m$. Indeed take the vertical crossings of $[0,S]\times [0,3S\mathrm{/}2]$ and $[0,S]\times [S\mathrm{/}2,2S]$, and a horizontal crossing of their overlap square $[0,S]\times [S\mathrm{/}2,3S\mathrm{/}2]$. The vertical bounds are [F4] after coordinate interchange; the shift $S\mathrm{/}2$ is even and preserves $P$. The square bound is also [F4]. Each horizontal path intersects both vertical paths by [F5], so their union gives the required vertical crossing. FKG [F2] gives $b^{2}a$. Translation by any integer vector with even coordinate sum preserves the lattice, the periodic FK law, and this estimate. Restricting a taller crossing at its first hit of any intermediate integer height gives a crossing of the truncated rectangle.

Suppose $n\ge 32$. Set $k=\lfloor n\mathrm{/}32\rfloor$, $L=16k$, $h=4k=L\mathrm{/}4$, $W=\lceil \alpha n\rceil$ and $J=\lceil W\mathrm{/}h\rceil$. For $0\le j\le J$ put $a_j=jh$, $b_j=-jh$, and take the closed squares $Q_j=[a_j,a_j+L]\times [b_j,b_j+L]\mathrm{.}$ For $0\le j<J$ take the closed connectors $C_j=[a_j+h,a_j+L]\times [b_j-h,b_j+L]\mathrm{.}$ Each connector has width $12k$ and height $20k$. Since $a_j+b_j=0$, every vertex of every square and connector has original coordinate $0\le y=(u+v)\mathrm{/}2\le L\le n$. The corner translations $(jh,-jh)$ preserve $P$. Every square embeds and has horizontal crossing probability at least $a$ by [F4], since $L<m$. A vertical aspect-two rectangle of the same connector width $S=12k$, starting at its bottom, has height $24k<n<m$. Step 1.2 bounds its vertical crossing probability below by $d$. Truncation at height $20k$ implies a vertical crossing of $C_j$, so that event too has probability at least $d$. Its witness may use the taller box to establish the numerical bound, but the event being intersected below is supported on $C_j$ itself.

Require horizontal crossings of all $Q_j$ and vertical crossings of all $C_j$. In $Q_j\cap C_j$ the horizontal path has a subpath crossing its width-$12k$ span, while the vertical path has a subpath crossing its full height-$L$ span. They intersect by [F5]. The same applies to $Q_{j+1}\cap C_j$, whose height-$L$ span is shifted down by $h$. Thus the union is a connected actual-open-edge set. A left endpoint of the $Q_0$ crossing has $u=0$ and hence $x\le 0$. A right endpoint of the $Q_J$ crossing has $u=Jh+L$ and $v\le -Jh+L$, hence $x\ge Jh\ge W$. Their joined path stays in $0\le y\le n$ by step 2.1. Truncate this path between its last vertex on $x=0$ before its first hit of $x=W$. It gives the required ordinary-grid crossing of $\{0,\dots ,W\}\times \{0,\dots ,n\}$. The patches need not jointly embed: lift the periodic configuration to the plane and choose the guaranteed crossings in these fixed lifts. Their adjacent overlaps are literal planar overlaps. The final target itself embeds, since $W<m$ and $n<m$, so its projected crossing is unambiguous.

By finite FKG [F2], the joint event in step 3.1 has probability at least $a^{J+1}{d}^J\ge \min (a,d{)}^{2J+1}$. Because $k\ge n\mathrm{/}64$ for $n\ge 32$, $h\ge n\mathrm{/}16$ and $h\ge 4$. Hence $J\le W\mathrm{/}h+1\le 16\alpha +5\mathrm{/}4$ and $2J+1\le 32\alpha +7\mathrm{/}2$. Its probability is at least $\min (a,d{)}^{\lceil 32\alpha +5\rceil }>0$, independently of $n,m$. When $1\le n<32$, use the fixed ordinary horizontal row of $W$ edges. All its labels are distinct since $W<m$. Repeated conditioning with [F2] gives probability at least ${\rho }^W\ge {\rho }^{\lceil 32\alpha \rceil }$. The minimum of these two positive constants is a permissible $c(\alpha ,q)$. This proves the stronger ceiling-width statement for all stated sizes.