Wired FK annuli and the numerical self-dual threshold

For an increasing event $B$ on a finite graph, differentiation of its finite FK weights gives ${\varphi }_p^{\mathrm{\prime }}(B)={\mathrm{Cov}}_p({\mathbf{1}}_B,\mathrm{\mid }\omega \mathrm{\mid })\mathrm{/}[p(1-p)]$. Writing $u_e={\varphi }_p(e\text{ open})$ and $I_B(e)={\varphi }_p(B\mid e\text{ open})-{\varphi }_p(B\mid e\text{ closed})$ yields ${\varphi }_p^{\mathrm{\prime }}(B)={\sum }_e{u}_e(1-u_e)I_B(e)\mathrm{/}[p(1-p)]$. By [F1] the coefficients have a positive lower bound on a compact parameter interval inside $(0,1)$. On an $L$-torus a translation-invariant $B$ has equal influences on each of the two edge orbits, each of size $L^2$. Thus [F4] implies ${\varphi }_p^{\mathrm{\prime }}(B)\ge c{\varphi }_p(B)(1-{\varphi }_p(B))\log L$. Integrating its logistic derivative between $p_s$ and fixed $p>p_s$ shows that ${\varphi }_{p_s}(B)\ge b>0$ implies $1-{\varphi }_p(B)\le C{L}^{-\eta }$, with constants independent of $L$.

We specify the finite stopping operation used for an outer circuit. In a finite ambient box list all simple circuits disjoint from ${\mathrm{\Lambda }}_{r_{k+1}}$, strictly inside ${\mathrm{\Lambda }}_{r_{k+2}}$, and surrounding the inner square. For a candidate $\gamma$, let $I(\gamma )$ be the edges whose relative interiors lie in its bounded component, including edges entering that component from circuit vertices. Let $O(\gamma )$ contain all other ambient edges, including circuit edges. The event selecting $\mathrm{\Gamma }=\gamma$ is: every edge of $\gamma$ is open and there is no surrounding open circuit in $O(\gamma )$ whose bounded interior strictly contains that of $\gamma$. These tests involve only $O(\gamma )$. To verify that exactly one candidate is selected whenever a surrounding circuit exists, splice any exterior excursion of another surrounding circuit with the outer arcs of $\gamma$, keeping the unbounded face on the outside, and erase subloops not surrounding the inner square. This gives a surrounding circuit in the union, on or outside $\gamma$, with strictly larger interior if that excursion protrudes. Applying this finite operation to all candidates yields an outermost circuit; two different maximal circuits would enlarge each other by the same surgery. Equal interiors have the same Jordan boundary. Thus the selected candidate is unique. For each output $(\gamma ,\tau )$, where $\tau$ is the full assignment on $O(\gamma )$, the output fiber is precisely the fixed-exterior cylinder $\{\omega {\mathrm{\mid }}_{O(\gamma )}=\tau \}$ if $\tau$ passes the tests, and is empty otherwise. Every interior assignment is admitted. This fiber identity, rather than a claim about a particular probing order, certifies that conditioning on the output never conditions an interior edge.

For $m\ge r$, reaching $\mathrm{\partial }{\mathrm{\Lambda }}_m$ entails $D_r$, and wired domain comparison gives $a_m\le {\varphi }_{m,p,q}^1(D_r)\le a_r$. In particular $a_n$ decreases, so ${\mathrm{\Theta }}^1={\inf }_n{a}_n$ exists. Taking $m\to \mathrm{\infty }$ in the displayed inequalities gives ${\mathrm{\Theta }}^1\le {\pi }_{p,q}^1(D_r)\le a_r$. Infimizing over $r$ proves ${\mathrm{\Theta }}^1={\inf }_r{\pi }_{p,q}^1(D_r)$. This is a pair of real inequalities, not an exchange of two limits.

On a finite edge set and for an increasing event $D$, its Hamming distance $H_D$ is the minimum number of closed edges to open to enter $D$. Opening one coordinate lowers it by zero or one; hence $\mathrm{\mid }\omega \mathrm{\mid }+H_D$ is increasing, $H_D$ is decreasing, and $H_D{\mathbf{1}}_D=0$. Finite differentiation and FKG give $\frac{d}{dp}{\varphi }_p(D)=\{{\mathrm{Cov}}_p({\mathbf{1}}_D,\mathrm{\mid }\omega \mathrm{\mid }+H_D)+{\varphi }_p(D){\mathbb{E}}_p{H}_D\}\mathrm{/}[p(1-p)]\ge 4{\varphi }_p(D){\mathbb{E}}_p{H}_D$. Integrate from $p$ to $t>p$ and use monotonicity of the decreasing $H_D$ to obtain ${\varphi }_p(D)\le \exp \{-4(t-p){\mathbb{E}}_t{H}_D\}$. Apply this to $D_n$ in wired ${\mathrm{\Lambda }}_m$ with $m>n$ and take $m\to \mathrm{\infty }$. Both sides involve only fixed finite cylinder sums, proving the claimed Hamming inequality.

We derive the exact numerical free/wired duality from a concrete free plane graph. Let $H_m$ be the free rectangular grid on the $(2m{)}^2$ unit-cell centers of ${\mathrm{\Lambda }}_m$. Each bounded face of $H_m$ surrounds one interior vertex of ${\mathrm{\Lambda }}_m$; its one unbounded face represents all vertices of $B_m$ identified together. A dual edge of $H_m$ crosses exactly one nonperimeter edge of ${\mathrm{\Lambda }}_m$, with precisely those interior-vertex or boundary-class endpoints. Thus $H_m^{\ast }$ is the wired quotient graph with the perimeter edge labels omitted. Each omitted perimeter edge is a loop after wiring, so its open/closed weight sums to $p+(1-p)=1$ independently and changes no retained-edge law. Apply [F5] to the connected free plane graph $H_m$ at parameter $t^{\ast }$, whose dual parameter is $t$. The inverse complement bijection sends the wired retained-edge law at $t$ to the free law on $H_m$ at $t^{\ast }$. This proves the finite identity before taking any limit; the exterior face is one of the wired-quotient vertices, not an additional cell-center vertex. The cell-center grids exhaust the translated dual lattice, and, after its fixed half-unit identification with ${\mathbb{Z}}^2$, lie between centered boxes with radii differing by a fixed constant. Finite domain comparison and [F2] therefore give, for every dual cylinder $E$, ${\pi }_{t,q}^1(\text{dual-complement }E)={\pi }_{t^{\ast },q}^0(E)$. This is a numerical cylinder identity, not a claim of an unconstructed infinite law.

To turn this estimate into a bound for one specified rectangle, first take width $2u$, height $2bu$ with integer $b\ge 1$, and torus side $L=16b^{2}u$. Let $B$ mean that some translate of a width-$u$, height-$L\mathrm{/}2$ rectangle has a vertical crossing. It is translation invariant and has self-dual probability bounded below by [F3], since its fixed aspect is $8b^2$ and its height is strictly less than $L$. Every such narrow strip is contained horizontally in one of the width-$2u$ strips starting on the $u$-grid, and its height interval contains a full height-$2bu$ band starting on the $2bu$-grid. Its crossing therefore crosses one of $M=(L\mathrm{/}u)(L\mathrm{/}(2bu))$ fixed translates of the target rectangle. Wrapping translates are lifted to the plane for this containment. If their crossing events are $B_i$, positive association of their decreasing complements gives $(1-{\varphi }_p(B_1){)}^M\le {\varphi }_p({\bigcap }_i{B}_i^c)\le {\varphi }_p(B^c)$. Step 1.1 proves ${\varphi }_p(B_1)\ge 1-C{u}^{-\eta \mathrm{/}M}$. The number $M$ is independent of $u$.

For a rectangle of width $w$ and height $h$ with $1\le h\mathrm{/}w\le \beta$, take $u=\lfloor w\mathrm{/}2\rfloor$ and a fixed integer $b$ depending only on $\beta$ with $2bu\ge h$. A crossing of the narrower, taller rectangle restricts to a crossing of the target. Rotating handles the other orientations. A fixed enlargement of $L\mathrm{/}u$ in step 2.1 preserves the proof and accommodates any surrounding planar box of size comparable to $w$. Condition that torus law on all edges outside the box; its induced partition is bounded above by the wired partition. Hence the wired box gives the specified rectangle a crossing probability at least $1-C{w}^{-\delta }$. The finitely many small widths are absorbed into $C$.

Fix $\alpha >1$. For large $k$ put $a=r_k+2$, $b=r_{k+1}-2$, $R=r_{k+2}$. Use vertical crossings of $[a,b]\times [-b,b]$ and $[-b,-a]\times [-b,b]$, horizontal crossings of $[-b,b]\times [a,b]$ and $[-b,b]\times [-b,-a]$, and a horizontal crossing of $[-R,R]\times [-b,b]$. In each corner square the adjacent side crossings intersect by [F6]; their union contains a surrounding closed walk, from which erasing nonsurrounding loops gives a simple circuit. The last crossing intersects the two vertical sides and connects that circuit to the outer boundary. All widths are comparable to ${\alpha }^k$ and all aspects are fixed. Step 3.1 and a union bound give ${\varphi }_{r_{k+2},p,q}^1(A_k)\ge 1-d_k$ with $d_k=C{e}^{-\epsilon k}$. Enlarge $K$ until $b>a$ and $d_k<1\mathrm{/}2$ for $k\ge K$.

Suppose $A_{k+1},\dots ,A_n$ hold. Apply step 1.2 to the next annulus. The selected $\mathrm{\Gamma }$ has an outgoing open connection to $\mathrm{\partial }{\mathrm{\Lambda }}_{r_{k+3}}$: an outgoing path from any surrounding circuit meets $\mathrm{\Gamma }$. These outer events are measurable in its returned exterior data. For $A_{k+1}$ one may use $\mathrm{\Gamma }$ itself, and any path excursion inside it can be replaced by an open circuit arc; circuits of $A_j$ for $j\ge k+2$ are outside it. On each output fiber, [F1] gives the interior FK law with all vertices of $\mathrm{\Gamma }$ wired together by its actual open edges. Let $G_k(\mathrm{\Gamma })$ mean that a circuit of the inner annulus connects to $\mathrm{\Gamma }$ inside its bounded domain. In wired ${\mathrm{\Lambda }}_{r_{k+2}}$, $A_k$ implies $G_k(\mathrm{\Gamma })$ by separation, even when the fixed geometric $\mathrm{\Gamma }$ is not open in that comparison law. Wired FK inside $\mathrm{\Gamma }$ dominates the restriction of wired FK on the larger square. Step 4.1 therefore gives conditional probability at least $1-d_k$ for $G_k(\mathrm{\Gamma })$. Its connection then follows the already exposed outgoing path through $\mathrm{\Gamma }$ to $\mathrm{\partial }{\mathrm{\Lambda }}_{r_{k+2}}$, giving $A_k$. Averaging the exterior fibers proves $P(A_k\mid A_{k+1},\dots ,A_n)\ge 1-d_k$. No square boundary is asserted to be wired in the random configuration.

Start in wired ${\mathrm{\Lambda }}_{r_{n+2}}$, where $A_n$ has probability at least $1-d_n$, and descend using step 5.1. This proves the finite product bound for ${\bigcap }_{k=N}^n{A}_k$, and hence for ${\bigcap }_{k=N}^m{A}_k$. For fixed $m$, take the deterministic outer-box limit along $n\to \mathrm{\infty }$; finite additivity and ${\prod }_{k=N}^n(1-d_k)\ge 1-{\sum }_{k=N}^n{d}_k$ prove the stated cylinder lower bound. No probability of an infinite intersection or measure extension is used.

For $p>p_s$ choose $N\ge K$ with ${\sum }_{k\ge N}{d}_k\le 1\mathrm{/}2$. Fix a deterministic nearest-neighbor tree containing all vertices of ${\mathrm{\Lambda }}_{r_{N+1}}$; if it has $M_0$ edges, its all-open event $T$ has probability at least ${\rho }^{M_0}$ by successive edge conditioning. Finite FKG and step 6.1 give ${\varphi }_{r_{n+2},p,q}^1(T\cap {\bigcap }_{k=N}^m{A}_k)\ge {\rho }^{M_0}\mathrm{/}2$ for $n\ge m\ge N$. Consecutive circuits connect because an outgoing path meets the next surrounding circuit. For any prescribed $R$, take the least $m\ge N$ with $r_{m+2}>R$; this finite chain and the tree imply $D_R$. Taking only the outer-box limit gives ${\pi }_{p,q}^1(D_R)\ge {\rho }^{M_0}\mathrm{/}2$ for every $R$, with the same constant. Step 1.3 proves ${\mathrm{\Theta }}^1(p,q)>0$.

Fix $0<s<p_s$ and choose one $t\in (s,p_s)$ such that $t^{\ast }\mathrm{\notin }{J}_q$. The preimage of $J_q$ under the explicit dual bijection is countable. A point outside it can be specified without Countable Choice: enumerate that exceptional set, start with a rational closed interval inside $(s,p_s)$, and recursively take the first smaller rational closed interval in a fixed enumeration which avoids the next exceptional point and has diameter at most half the previous diameter. Its unique limit lies outside the exceptional set. Since $t^{\ast }>p_s$, step 6.1 at the dual parameter, for any chosen $\alpha >1$, supplies $N$ such that every finite intersection of dual annulus events with indices from $N$ onward has numerical wired weight at least $1\mathrm{/}2$. By equality of the free/wired cylinder weights at $t^{\ast }$ and step 1.5, the same finite dual-circuit intersections have primal wired cylinder weight at least $1\mathrm{/}2$ at $t$.

Retain every other annulus, so its dual circuits have disjoint edge supports. Among the annuli strictly inside ${\mathrm{\Lambda }}_{n-2}$ there are at least $(\log n)\mathrm{/}(2\log \alpha )-O(N+1)$ retained circuits; the fixed dual-lattice translation changes only the cutoff. Every primal path realizing $D_n$ must cross each such circuit at a distinct closed primal edge. On the finite cylinder supporting these circuits and $H_n$, multiply this pointwise inequality by its nonnegative atom weights and sum. Step 7.2 gives ${\mathbb{E}}_{{\pi }_{t,q}^1}{H}_n\ge (\log n)\mathrm{/}(8\log \alpha )$ for large $n$. Taking $\log \alpha =(t-s)\mathrm{/}(4d+8)$ in step 1.4 gives ${\pi }_{s,q}^1(D_n)\le n^{-(2d+4)}$ for sufficiently large $n$, proving the claimed bound after enlarging $C_d$. In particular these probabilities tend to zero, so step 1.3 yields ${\mathrm{\Theta }}^1(s,q)=0$. All intersections and expectations used here are finite.

Every parameter strictly above $p_s$ belongs to the set defining the numerical threshold by step 7.1, and none strictly below belongs by step 8.1. Its infimum is therefore $p_s$. The proof never assigns a value at $p_s$ and never invokes an infinite-volume probability measure or Countable Choice.