Model and statement

Let \(\rho\) be a probability measure supported on \([-1,1]\) such that \[ \int x\,\rho(\mathrm dx)=0, \qquad \sigma_\rho^2:=\int x^2\,\rho(\mathrm dx)>0. \tag{1}\] On the unoriented nearest-neighbor edges of \(\mathbb Z^2\), let \((\xi_e)_e\) be independent with law \(\rho\), and set \[ J_e=1+\varepsilon\xi_e,\qquad 0<\varepsilon<1. \tag{2}\] Write \(\omega=(\xi_e)_e\) for the environment and \(\mathbb P_\rho\) for its product law. For a finite spin domain \(\Lambda\), inverse temperature \(\beta\geq0\), and prescribed exterior spins \(\tau\), the quenched Ising law has weight \[\exp\!\left(\beta\sum_{\{x,y\}:\{x,y\}\cap\Lambda\ne\varnothing} J_{\{x,y\}}\sigma_x\sigma_y\right), \qquad \sigma_x=\tau_x\quad(x\notin\Lambda),\] where the sum runs over nearest-neighbor edges with at least one free endpoint. Expectations for this law are denoted by \(\langle\,\cdot\,\rangle^\tau_{ \Lambda,\beta,J}\). Put \(B_m=[-m,m]^2\cap\mathbb Z^2\) and define \[ \beta_c^\rho(\varepsilon) =\inf\left\{\beta\geq0: \mathbb E_\rho\!\left[\lim_{m\to\infty} \langle\sigma_0\rangle^+_{B_m,\beta,J}\right]>0\right\}. \tag{3}\] The plus-state limit exists by ferromagnetic monotonicity. Throughout the paper, averaging a quenched law means integrating it against \(\mathbb P_\rho\); the environment is never reweighted by a partition function. We write \(K_0=\tfrac12\log(1+\sqrt2)\) for the pure critical coupling.

Let \(D\subset\mathbb C\) be a bounded Jordan domain with distinct marked boundary points \(a,b\). A lattice approximation is a simply connected square-lattice domain containing every interior nearest-neighbor bond, embedded with mesh \(\delta\), with a Jordan boundary and marked interface endpoints \(a_\delta,b_\delta\). We call such a domain full. The boundary approximation used here means that there are homeomorphisms \[ h_n:\partial D\longrightarrow\partial D_n, \qquad \sup_{z\in\partial D}|h_n(z)-z|\longrightarrow0, \qquad (a_n,b_n)\longrightarrow(a,b). \tag{4}\] The graphs and bond variables live on the unit lattice and are then embedded by multiplication by \(\delta_n\). This convention allows all \(n\) to use one environment. The exterior-neighbor and pinned-boundary-vertex conventions are specified in Section 7.

Dobrushin conditions fix plus spins on one marked arc and minus spins on the other. The arc assignments must give a consistent spin at every pinned site. The spin interface is oriented from \(a_n\) to \(b_n\) and uses the deterministic north-east/south-west pairing whenever a contour vertex has degree four. We use continuous polygonal representatives of the resulting oriented curves. For two such curves, let \[ d_{\rm curv}(\gamma,\eta) =\inf_{\phi,\psi}\sup_{t\in[0,1]} |\gamma(\phi(t))-\eta(\psi(t))|, \qquad \bar d_{\rm curv}=1\wedge d_{\rm curv}, \tag{5}\] where \(\phi,\psi\) range over increasing homeomorphisms of \([0,1]\); curves at zero distance are identified. For probability laws on this space, set \[ d_{\rm BL}(\mu,\nu) =\sup\left\{\left|\int F\,\mathrm d\mu-\int F\,\mathrm d\nu\right|: \|F\|_\infty\leq1,\ |F(\gamma)-F(\eta)|\leq\bar d_{\rm curv}(\gamma,\eta)\right\}. \tag{6}\] This is a bounded-Lipschitz metric for weak convergence in the oriented uniform-curve topology. Write \(Q_{n,\omega}\) for the quenched Dobrushin interface law and \(\mathsf S_{D;a,b}\) for chordal \(\mathop{\mathrm{SLE}}_3(D;a,b)\).

History and relation to earlier work

The effect of weak bond disorder is especially delicate in two dimensions. Harris’s relevance criterion places the pure Ising model at the marginal case, where the specific-heat exponent is zero and the criterion alone does not decide the effect of randomness [15]. Dotsenko and Dotsenko’s fermionic analysis predicted a double logarithmic specific-heat singularity [7, 8]. Renormalization-group work of Jug, Shalaev, Shankar, and Ludwig developed and refined the description of logarithmic corrections to pure-model critical behavior [16, 25, 26, 19]. The predictions for different correlation observables were not identical throughout this development. The relevant physical picture here is marginal irrelevance: weak disorder decreases under changes of scale, but only logarithmically. These works concern thermodynamic quantities and correlations; control of a quenched law for an entire interface also requires geometric estimates.

For the homogeneous critical model, the conformal description is rigorous. Schramm’s stochastic Loewner evolution supplies the candidate interface laws [24]. Smirnov established conformal convergence of the FK-Ising fermionic observable [28], and Chelkak and Smirnov proved fermionic-observable universality for critical isoradial Ising models [6]. Chelkak, Duminil-Copin, Hongler, Kemppainen, and Smirnov proved that the critical square-lattice spin interface with Dobrushin boundary conditions converges to chordal \(\mathrm{SLE}_3\) in uniform distance modulo increasing reparametrization [5]. This is the pure interface theorem underlying our comparison argument.

Rigorous universality results for deterministic interactions provide a second point of comparison. Giuliani, Greenblatt, and Mastropietro proved universality of infinite-plane energy-correlation scaling limits for weak finite-range pair perturbations with lattice symmetries [13]. For non-integrable spin interfaces, Greenblatt and Peltola constructed a Grassmann representation of a discrete martingale and formulated the correlation estimates needed for a proposed route to \(\mathrm{SLE}_3\) [14]. Their paper does not prove interface convergence. For random bonds, Mahfouf proved quenched FK-interface convergence to \(\mathrm{SLE}_{16/3}\) in a regime where the disorder strength tends to zero with the mesh [20]. Avérous and Mahfouf established high-probability crossing estimates for near-critical random-bond FK models, also with vanishing disorder for \(1<q\le4\) [2]. In the present theorem the nonzero disorder strength is fixed as the mesh tends to zero, and the interface separates spin clusters.

The main theorem of [22] treats symmetric two-valued bond disorder, whereas its local scale transformation, moment estimates, and marginal variance estimate are formulated for more general random local potentials. We verify their hypotheses for the law \(\rho\) and its exact dual, and use the pure and deterministic boundary-comparison results of [23] through their formulations in [22]. The critical-point argument adapts the torus method of [22] to match two separately tuned trajectories within the same physical temperature family. This matching replaces distributional self-duality.

Duality has a substantial history in the study of disordered critical points. Chayes and Shtengel established vanishing magnetization and critical correlation properties for self-dual disordered ferromagnets under their stated hypotheses [4]. Our general one-edge law need not be self-dual. We use Kramers–Wannier duality [18] in the finite-torus form of Bugrij and Shadura, which relates the four periodic/antiperiodic partition functions for arbitrary positive inhomogeneous couplings [3]. The torus comparison identifies the two tuned temperatures; a separate argument identifies their common value with the spontaneous-magnetization threshold.

The remaining argument combines classical probabilistic tools with the geometric inputs of [22, 23]. The Fortuin–Kasteleyn representation [11], the Edwards–Sokal coupling [10], and the FKG lattice inequality [12] relate spin barriers, random-cluster connections, and boundary monotonicity. The decision-tree inequality of O’Donnell, Saks, Schramm, and Servedio [21], extended to monotonic measures by Duminil-Copin, Raoufi, and Tassion [9], is applied at each fixed environment before averaging over the bonds. The geometric use of crossing estimates to control random curves has precedents in Aizenman–Burchard [1] and Kemppainen–Smirnov [17]; Sheffield–Sun [27] distinguish convergence of driving functions from convergence in the uniform curve topology. The specific passage from boundary barriers and trace convergence to oriented-curve convergence used here is the progress criterion of [22].

The extension and the proof

The local scale transformation of [22] rewrites the model as a sequence of random local perturbations of pure critical Ising. Section 2 states the analytic estimates, and Section 3 verifies their hypotheses for the primal and exact dual families and chooses a temperature for each at which the perturbations tend to zero in environmental moments. The variance coordinate retains local cubic terms, so the estimates also cover laws with nonzero third moment.

The two families are parametrized by the same physical inverse temperature, but their selected temperatures need not initially agree. Section 4 compares the four torus partition functions along these trajectories. Exact duality shows that, at both selected temperatures, the same primal FK winding event has probability asymptotic to its pure critical probability. Monotonicity and a quantitative lower bound for its temperature derivative then force the temperatures to agree. This identifies a common primal-dual temperature without assuming symmetry of the disorder law.

Section 5 transfers the decaying perturbations to local conditional crossing estimates and joint laws of finitely many crossing tests on tori. The error terms are geometrically weighted; decay suffices even though the scale norms need not be summable. Section 6 uses the estimates on both lattices to identify the common temperature with (3). Barriers in annuli with disjoint bond neighborhoods first rule out percolation there. Above that temperature, an environmental amplification argument and the decision-tree inequality give rapid dual-arm decay and positive primal magnetization.

Section 7 uses the boundary-comparison results of [22, 23] to compare the random interface with pure interfaces in slightly displaced domains. The couplings retain the pure comparison law conditional on the original environment. This conditional marginal turns the comparison into a quenched statement. A planar sandwich identifies the trace, and a separate multi-passage estimate controls its order of traversal, giving the full metric in (5).

The reference field and its energy charge

Let \(\mathbb E_0\) denote expectation under the full-plane critical Ising law at \(K_0\), together with a locally finite family of independent, finite-valued reference variables attached to specified locations. These variables are part of the Gibbs field: they are integrated or conditioned along with the spins. Spin reversal leaves them unchanged. Their product reference law, including their locations and labels, respects lattice translations and square symmetries. At the initial scale there are no such variables.

For an edge \(e=\{x,y\}\), put \[b_e=\sigma_x\sigma_y-\mathbb E_0(\sigma_x\sigma_y), \qquad B_x=\frac14\sum_{e\ni x}b_e.\] The theorem “Exact pure energy charge” in [22] provides the linear functional \[ Q(f)=\kappa^{-1}\lim_{|x|\to\infty}|x|^2\mathbb E_0(fB_x), \qquad \kappa=\pi^{-2}, \tag{8}\] on bounded, spin-flip-even local observables. The reference variables are included in this expectation. Thus \(Q\) measures the energy component of a local perturbation through its correlation with a distant energy observable. The functional kills constants, is invariant under lattice translations and square symmetries, and satisfies \(Q(b_e)=1\). Pure conditional integration across a surrounding square preserves \(Q\) exactly. If the support of \(f\) lies in a square of radius \(r\), then \[ |Q(f)|\le C r\|f\|_\infty. \tag{9}\] This normalization permits charges at successive scales to be compared without accumulating a normalization error.

Arrays and the exact scale transformation

At an integer spacing \(s\), a local array is a family \(f=(f_i)_{i\in\mathbb Z^2}\) of random functions based at the cell centers \(si\). We impose the following hypotheses. For every environment, each \(f_i\) is real, bounded, spin-flip even, and pure centered: \(\mathbb E_0 f_i=0\). Both its field support and the primitive environmental variables determining it lie within distance \(Cs\) of \(si\), in supremum norm, for the fixed support constant of the scale construction. The primitive environmental variables are independent. The law of the array is invariant under translations of the \(s\)-grid and the symmetries of a square, acting also on reference-variable locations and labels. These are symmetries in distribution; an individual array need not have them. Finally, \(\mathbb E\|f_0\|_\infty^{40}<\infty\).

Here and throughout the scale argument, \(\mathbb E\) averages the bond environment and the independent priorities and thresholds used to construct the transformation. These choices are held fixed when sampling the Gibbs field. They are distinct from the reference variables averaged by \(\mathbb E_0\). In a finite volume, the array adds \(\sum_i f_i\) to the logarithm of the reference density, with all old exterior field variables prescribed. The sum includes the potentials meeting the volume.

The proposition “The local scale map” in [22] gives, for a sufficiently large odd integer \(L\) and a threshold parameter \(b>0\), a map from such arrays at spacing \(s\) to arrays at spacing \(s'=Ls\) with the same support constant. The construction uses independent continuous priorities and independent environmental thresholds uniform on \([b,2b]\) to determine its local choices and to flag large input terms. The map introduces normalized conditional likelihoods for fresh reference coins, then uses conditional integrations in separated squares to replace local logarithmic interactions. These are changes of the interaction functions: the enlarged field alphabet retains every old spin and reference variable. Centering the resulting potentials changes only a scalar factor. For each fixed environment and arbitrary old exterior assignment, these operations preserve the exterior-integrated weight exactly. The same formulas and full-plane centering constants apply in embedded torus charts. In particular, the construction is compatible with a flat spin twist: a change of trivialization in a local chart is a constant spin flip, under which all potentials are invariant.

We will use three quantitative and local features of that proposition, the corollary “Local implementation”, and the proposition “A coupling of consecutive fields” in the same section. There are fixed numerical constants \(K,K_1\) and a constant \(C_L\), depending on the fixed step parameters, such that \[ \|f'_j\|_\infty\le C_L \sum_{i:\,\|si-s'j\|_\infty\le Ks'}\|f_i\|_\infty. \tag{10}\] In a finite free chart with prescribed old exterior field data, the transformation agrees with the full output specification after erosion by \(K_1s'\). This assertion includes annular charts. Every fresh reference variable with a nonconstant likelihood is sampled as part of the construction. Exactness does not assert the same input kernel if fresh coins are independently prescribed before the construction; disintegration over the sampled output exterior remains available. Finally, consecutive fields can be coupled to agree outside squares of radius \(5s'\) about marked parent centers. Conditional on the enlarged environment, these marks are independent of one another and of the output Gibbs sample, with local parameters satisfying \[ p_j\le \min\left\{1,C_L \sum_{i:\,\|si-s'j\|_\infty\le Ks'}\|f_i\|_\infty\right\}, \qquad \mathbb Ep_j\le C_L(\mathbb E\|f_0\|_\infty^{40})^{1/40}. \tag{11}\] Both the coupling and these bounds are uniform over the prescribed old exterior assignments. These exactness and locality statements will later connect the moment estimates to crossing events.

Moment and variance coordinates

Write \(q_s=Q/s\), and define \[ \begin{aligned} X&=(\mathbb E\|f_0\|_\infty^{40})^{1/40}, &Y&=\|\mathbb Ef_0\|_\infty,\\ m&=\mathbb Eq_s(f_0), &W&=\sum_{i\in\mathbb Z^2}\mathop{\mathrm{Cov}}\bigl(q_s(f_0),q_s(f_i)\bigr). \end{aligned} \tag{12}\] All covariances here are environmental. The covariance sum is finite by the dependence hypothesis. It is nonnegative because, for growing squares \(\Lambda\subset\mathbb Z^2\), \[W=\lim_{\Lambda\uparrow\mathbb Z^2}\frac1{|\Lambda|} \mathop{\mathrm{Var}}\left(\sum_{i\in\Lambda}q_s(f_i)\right).\] The mean charge \(m\) is the expanding coordinate of the map. The variance density \(W\) is marginal at its linearization. To see its decay, one needs a coordinate that also retains local cubic and quartic terms.

The construction in [22] begins with a finite random insertion \(F=\sum_{i\in I}t_i f_i\), where \(I\subset\mathbb Z^2\) is finite and the \(t_i\) are real parameters. Its normalized charge is \[T(F)=q_s\left(\frac{e^F}{\mathbb E_0e^F}\right).\] We take Taylor coefficients at zero for each fixed environment; the expansions below use only finite moments, not environmental exponential integrability. Since \(\mathbb E_0F=0\) and \(q_s\) kills constants, the first three homogeneous terms are \[T_1=q_s(F),\qquad T_2=\tfrac12q_s(F^2),\qquad T_3=\tfrac16q_s(F^3)-\tfrac12q_s(F)\mathbb E_0F^2.\] The terms of degrees two, three, and four in the formal environmental variance of \(T(F)\) are therefore \[\mathop{\mathrm{Var}}(T_1),\qquad 2\mathop{\mathrm{Cov}}(T_1,T_2),\qquad \mathop{\mathrm{Var}}(T_2)+2\mathop{\mathrm{Cov}}(T_1,T_3).\] The cubic term need not vanish for a nonsymmetric law. We retain it while estimating the degree-four drift: an uncontrolled cubic error would be larger than the quartic change we seek to determine.

The coordinate \(v_H\) is this variance polynomial through degree four, per cell, with a cutoff on the diameter of the input labels. More precisely, choose a dependence range \(R_0\) in cell units so that sets separated by more than \(R_0\) use disjoint environmental variables and have disjoint declared supports. For an ordered tuple \(\boldsymbol i=(i_1,\ldots,i_d)\), \(2\le d\le4\), give each occurrence its own indeterminate, including repeated indices, and set \[ C_d(\boldsymbol i)= \left.\partial_{t_1}\cdots\partial_{t_d}\, \mathop{\mathrm{Var}}\left[ q_s\left( \frac{\exp(\sum_{a=1}^d t_a f_{i_a})} {\mathbb E_0\exp(\sum_{a=1}^d t_a f_{i_a})} \right)\right]\right|_{\boldsymbol t=0}. \tag{13}\] The derivatives mean coefficient extraction from the displayed variance polynomial. For \(H>R_0\), define \[ v_H=\sum_{d=2}^4\frac1{d!} \sum_{i_2,\ldots,i_d\in\mathbb Z^2} \mathbf 1_{\{\mathop{\mathrm{diam}}(0,i_2,\ldots,i_d)\le H\}} C_d(0,i_2,\ldots,i_d), \tag{14}\] where diameter is Euclidean diameter in cell units. This is a real polynomial coordinate, not necessarily positive on an arbitrary array. Its quadratic part is exactly \(W\).

These are respectively the propositions “One-step estimates” and “Marginal variance flow” of [22], with the near-\(W\) estimate included in the latter. Their hypotheses are the array hypotheses above. The required symmetry is spatial symmetry in law, together with spin-flip evenness of the potentials; there is no assumption of invariance under negating the bond randomness. In the variance calculation, independence eliminates isolated centered environmental occurrences, while the cubic terms in (14) are kept. Thus an asymmetric law does not require any alteration of the imported estimates. The next section verifies their hypotheses for both temperature families and uses the negative term in (17) to keep the entire iteration small.

Primal and dual initial arrays

Write \(K_e(\beta)=\beta J_e=\beta(1+\varepsilon\xi_e)\) for the coupling including inverse temperature. For \(K>0\), define its dual coupling \(g(K)\) by \[ e^{2g(K)}-1=\frac{2}{e^{2K}-1}. \tag{18}\] Thus \(g\) is a smooth decreasing involution, \(g(K_0)=K_0\), \(g'(K_0)=-1\), and \(g''(K_0)=2\sqrt2\). The edge odds and their duals are \[w_e(\beta)=e^{2K_e(\beta)}-1, \qquad w^*_{e^*}(\beta)=\frac2{w_e(\beta)}, \qquad K^*_{e^*}(\beta)=g(K_e(\beta)),\] where \(e^*\) crosses \(e\). We identify the dual square lattice with a translate of the square lattice.

Use the scale-step parameters of Proposition 2 and choose an integer starting spacing \(s_0\) sufficiently large for the construction. We also take it large enough for the fixed geometric checks used later in the crossing estimates of [22]. Both lattices use the spacings \(s_k=s_0L^k\). The two input families, parametrized by a small number \(h\), are \[ \begin{array}{c|c|c} \text{family}&\text{physical inverse temperature} &\text{couplings supplied to the scale map}\\ \hline \mathrm p&K_0+h&(K_0+h)(1+\varepsilon\xi_e)\\ \mathrm d&K_0-h&g((K_0-h)(1+\varepsilon\xi_e)). \end{array} \tag{19}\] The opposite sign of \(h\) in the second row makes the mean charge increase with \(h\) in both initializations. No symmetry of the distribution is involved in this parametrization.

Write \(\widetilde K_e\) for the coupling in either row, on the lattice of that row. Allocate each edge to its nearest \(s_0\)-grid center, with distance measured from its midpoint and ties split equally. If \(\alpha_{i,e}\) is its allocation to \(s_0i\), then \(\alpha_{i,e}\ge0\), \(\sum_i\alpha_{i,e}=1\), and the allocation respects grid translations and square symmetries. Define \[ f_i^{(0)}=\sum_e\alpha_{i,e}(\widetilde K_e-K_0)b_e. \tag{20}\] The sum of these functions is precisely the change from the pure Ising logarithmic weight, up to a spin-independent constant. This remains true for conditional specifications with exterior pins, including the bonds incident to the free spins.

Each initial potential is bounded, even, pure centered, and determined by bonds in a fixed multiple of its cell. The iid bond law and the equivariant allocation give grid stationarity and square symmetry in law. For the dual array, the crossing-edge correspondence is a bijection, so the transformed bonds are again iid on the dual lattice. All the array hypotheses of Section 2 therefore hold for both families. They persist under iteration. Moreover, boundedness of the initial disorder and (10) give a deterministic bound on each potential at every fixed number of steps, uniformly on a compact interval of \(h\).

Recall \(\sigma_\rho^2=\mathbb E_\rho\xi^2>0\), and put \[C_\rho=2K_0^2\sigma_\rho^2.\] The strict positivity is where nondegeneracy of \(\rho\) enters the tuning. The following estimates also locate both temperatures to second order in the disorder strength.

Keeping the expanding coordinate between two walls

We now adapt the shooting argument of [22] to these two initializations. The purpose of the argument is to control the expanding mean charge while the variance slowly decreases. Its continuity step must accommodate atoms in \(\rho\), and will use the independent continuous thresholds of the scale construction.

The shooting construction alone does not establish uniqueness. Each selected temperature gives an exact small-potential representation at every scale, with the perturbation vanishing in environmental moments. The next argument compares these representations on a common torus and forces their physical temperatures to agree.

FK differentiation at fixed environment

For a finite graph \(G=(V,E)\) and a partition \(\pi\) of its boundary vertices, let \[ \phi_{G,\beta,\omega}^{\pi}(\eta) =\frac{1}{\mathcal Z_{G,\beta,\omega}^{\pi}} 2^{k_\pi(\eta)}\prod_{e\in E}w_e(\beta)^{\eta_e}, \qquad \eta\in\{0,1\}^{E}, \tag{33}\] where \(w_e(\beta)=e^{2\beta J_e}-1\) and \(k_\pi\) counts open clusters after the boundary identifications. The superscript \(\mathrm w\) denotes full wiring. All connection events below use ordinary edges; boundary identifications are not steps in a path. At fixed environment, these measures are monotonic measures: every edge’s conditional open probability increases when more of the other edges are open. They satisfy positive association, increase with the edge odds and the boundary wiring, and have the domain Markov property [11, 12]. Edwards–Sokal coloring couples them to the Ising law [10]. The dual measure, denoted by an additional superscript \(*\), has odds \(w^*_{e^*}(\beta)=2/w_e(\beta)\) as in (18). The parameter \(\beta\) in this notation is always the physical primal inverse temperature.

Two finite-volume formulas will be used both here and in Section 6. For an increasing event \(A\), set \(q(\beta)=\phi_{G,\beta,\omega}^{\pi}(A)\). Differentiating the finite sum in (33) gives \[ q'(\beta)=\sum_{e\in E}a_e(\beta) \mathop{\mathrm{Cov}}_{\phi_{G,\beta,\omega}^{\pi}}(\mathbf 1_A,\eta_e), \qquad a_e(\beta)=\frac{\mathrm d}{\mathrm d\beta}\log w_e(\beta) =\frac{2J_e}{1-e^{-2\beta J_e}}. \tag{34}\] On every compact interval in \((0,\infty)\) these coefficients have uniform positive lower bounds, because \(1-\varepsilon\le J_e\le1+\varepsilon\). For the dual law the same identity holds with coefficient \(-a_e(\beta)\) at the crossing edge. All the covariances are nonnegative.

If a decision tree determines \(A\) and queries edge \(e\) with probability \(\delta_e\), the decision-tree inequality for monotonic measures gives \[ \phi(A)(1-\phi(A)) \le C_{\rm DT}\sum_{e\in E}\delta_e\mathop{\mathrm{Cov}}_\phi(\mathbf 1_A,\eta_e). \tag{35}\] Here \(C_{\rm DT}\) is an absolute constant; an independent randomized choice of tree is allowed. We use the monotonic-measure extension [9] of the OSSS inequality [21]. Both (34) and (35) are applied at fixed environment, before any environmental expectation.

Four sectors on a torus

Torus duality mixes partition functions with different spin twists, so we retain all four sectors. Normalizing their vector removes a common bulk factor introduced by the scale transformations. The resulting comparison will make the primal FK winding probabilities at the two tuned temperatures asymptotically equal. The derivative bound will then force those temperatures to agree.

Let \(\mathbb T_N=(\mathbb Z/N\mathbb Z)^2\), \(N\ge3\). A flat spin twist with holonomy \(\alpha=(\alpha_1,\alpha_2)\in\{0,1\}^2\) is an assignment of edge signs \(\tau_e^\alpha\) whose product around every elementary face is \(1\) and whose products around the two coordinate cycles are \((-1)^{\alpha_1}\) and \((-1)^{\alpha_2}\). Define \[Z_\alpha(K)=\sum_{\sigma\in\{-1,1\}^{\mathbb T_N}} \exp\left(\sum_{e=\{x,y\}}K_e\tau_e^\alpha\sigma_x\sigma_y\right).\] A change of the twist by a vertex gauge transformation is absorbed by changing the spin variables. Thus \(Z_\alpha\) depends only on the holonomy. Order the four entries as \(00,10,01,11\), and write \(Z(K)=(Z_\alpha(K))_\alpha\) and \(\widehat Z(K)=Z(K)/\|Z(K)\|_2\).

Buffered tests and the three passage estimates

Draw the northwest–southeast diagonal in each lattice square and interpolate spins affinely. The zero contours have the prescribed north-east/south-west pairing, up to a displacement of one mesh unit. A colored path uses vertices with the specified sign and edges of this triangulation; the diagonals introduce no additional Ising interactions. A corridor is a polygonal topological rectangle with two opposite marked boundary arcs, called gates. Its crossing test asks for a colored path between the gates. A band is the closed region between two disjoint simple polygons; its circuit test asks for a colored path separating them. We approximate these fixed nondegenerate polygons by subcomplexes, preserving the order of marked sides, with an error of order one lattice spacing. Each test comes with a specified open neighborhood in which spins, and the incident lattice edges, remain free. This neighborhood is its buffer.

A passage between two terminal sets is a contour subarc joining them, with interior in the prescribed region. Distinct passages have disjoint open parameter intervals on each contour, although endpoints at terminal hits may coincide. Thus several passages of one contour are counted separately. Cropping at the last hit of the first terminal before the first hit of the second gives precisely this convention.

For \(S_r(z,\alpha)=z+e^{i\alpha}[-r,r]^2\), consider the annulus \(A(z,\alpha;r,R)=S_R(z,\alpha)\setminus\operatorname{int}S_r(z,\alpha)\). Write \(F_4\) and \(F_1\) for the events of at least four and at least one passage across this annulus. Write \(H_2\) for two passages across it confined to \(z+e^{i\alpha}\{\operatorname{Im}w\geq0\}\). The line in this last definition is a geometric constraint inside a free bulk chart; it carries no boundary pins.

We use the pure inputs of [23]. For some \(c_0>0\), they give \[ \begin{aligned} \limsup_{\delta\downarrow0}\mathbb P_0(F_4(r,R))&\leq C(r/R)^{2+c_0},\\ \limsup_{\delta\downarrow0}\mathbb P_0(F_1(r,R))&\leq C(r/R)^{c_0},\\ \limsup_{\delta\downarrow0}\mathbb P_0(H_2(r,R))&=o(r/R). \end{aligned} \tag{44}\] The ratio \(r/R\) is fixed before the mesh limit. Conditional laws with arbitrary pins beyond proportional inner and outer free collars have density at most a fixed factor \(D\) relative to the pure bulk marginal; \(D\) does not grow with \(R/r\). These assertions permit the fixed crops, rotations, and translations needed below.

Fix a gap factor \(G\) that separates successive ring collars, including their enlarged environmental neighborhoods. Choose smaller collars that will survive the local scale changes. For a candidate ratio \(a>1\) and each shape \(i=4,h,1\), let \(q_i^0\) be a deterministic upper bound for its pure conditional probability at ratio \(a\), uniform over placements and arbitrary pins beyond these smaller collars, once its inner lattice radius is sufficiently large. These bounds include the conditional density factor and the fixed crop losses. The pure estimates allow us to choose a large fixed \(a\), such bounds \(q_i^0\), exponents, and tolerances satisfying \[ p_4>2,\qquad p_h>1,\qquad p_1>0,\qquad q_i^0+\tau_i<(Ga)^{-p_i},\quad \tau_i>0,\quad i=4,h,1. \tag{45}\] For \(H_2\), first obtain strict slack below \((Ga)^{-1}\) from the little-oh bound and then choose \(p_h>1\) sufficiently close to one. Require \(s_0\) large enough that these pure bounds hold at finite lattice scales for the three fixed shapes; this is the additional starting-scale choice reserved in Section 3. Slightly relaxed radii and compactness of directions and normalized offsets give uniformity over their placements. All choices in this paragraph precede the choice of the disorder interval.

One deterministic input explains why exactly these three exponents occur. By [23], if changing spins in a square of radius \(b\) about \(z\) changes a test \(T\), the common exterior contours force successive four-passage, half-plane two-passage, and one-passage checks. More precisely, let \(a_T>0\) be smaller than the relevant side lengths and separations, and put \[ u=\min\{a_T,\max\{b,\mathop{\mathrm{dist}}(z,\mathcal E_T)\}\},\qquad v=\min\{a_T,\max\{b,\mathop{\mathrm{dist}}(z,\mathcal V_T)\}\}, \tag{46}\] where \(\mathcal E_T\) and \(\mathcal V_T\) are its polygon sides and vertices, including confinement walls for \(H_2\). The three checks are forced between scales \(b,u\), then \(u,v\), then \(v,a_T\). A bounded number of rings may be omitted at transitions, and their centers may move to the nearby side or vertex. The implication permits arbitrary fillings of the changed square. For the three fixed annular shapes, choose \(a_T\) as a sufficiently small multiple of the inner radius \(r\), so every retained ring has inner radius at most \(r/2\) and its collars stay in the free buffer.

Figure 1 summarizes these ranges. The scales \(u\) and \(v\) locate the change from a two-dimensional set of possible edit centers to neighborhoods of sides and then vertices.

Local conditional laws and a single reversal

Here and until physical lengths are explicitly introduced, distances are in lattice units. If \(W\) is a free chart and \(\zeta\) specifies the exterior spin and reference auxiliary variables, the level-\(k\) specification is \[ \mu_{k,\eta,W}^{\zeta}(\mathrm d\chi_W) =\frac{\exp F_{k,W}^{\zeta}(\chi_W)} {\mu_{{\rm pure},W}^{\zeta}(\exp F_{k,W}^{\zeta})} \mu_{{\rm pure},W}^{\zeta}(\mathrm d\chi_W), \qquad F_{k,W}^{\zeta}=\sum_{i:\,\mathop{\mathrm{supp}}f_i^{(k)}\cap W\ne\varnothing} f_i^{(k)}(\chi_W\zeta_{W^c}). \tag{47}\] The reference law is pure critical Ising times the product law of its auxiliary variables. At level zero, \(\mu_{0,\eta,W}^{\zeta}\) is the conditional nearest-neighbor Ising law under consideration.

We import the “Local implementation” and “A coupling of consecutive fields” of [22]. Their hypotheses are the array hypotheses used in the scale construction: each potential is bounded, real, spin-flip even and pure centered, with field support and primitive environmental dependence within \(Cs_k\) of its cell; the primitive variables are independent; and the array is stationary on the scale grid and square symmetric in law. A finite first moment suffices for the mark estimate below, and follows here from the fortieth moment. No symmetry under negating a bond variable is required.

The local implementation has the following precise conditional scope. With old exterior field variables fixed, the step can be performed inside \(W\) and is exact. Nontrivial fresh-coin decorations whose full stencils leave \(W\) are suppressed, their old logarithmic terms are retained, and integrations whose full stencils leave \(W\) are skipped. All pairwise coin comparisons used to select separated parent cells are retained, including comparisons that do not lead to an integration; retained logarithmic terms are included in each permitted integration they meet. Every nontrivially decorated fresh coin is therefore sampled inside \(W\). The output specification agrees with the full level-\((k+1)\) specification at distance at least \(C_1s_{k+1}\) from \(\partial W\). Conditioning on the exterior of this eroded chart after sampling the output gives a mixture of its conditional specifications. Exactness does not assert an unchanged input kernel if the fresh coins are independently pinned beforehand.

For fixed \(\eta\), reversal changes spins only in marked squares of radius \(b_k=5s_{k+1}\) about parent centers. The marks are independent conditional on \(\eta\) and independent of the entire output Gibbs sample. Choose the parameters themselves as exterior-independent measurable functions of the local primitive-variable stencil, with \[ p_{x,k}(\eta)\leq\min\left\{1, C_L\sum_{i:\,|s_ki-x|_\infty\leq C_Ls_{k+1}}\|f_i^{(k)}\|\right\}, \qquad \mathbb Ep_{x,k}\leq C_LX_k. \tag{48}\] This applies also to the localized step and to unused candidate centers. For example, if \(S_x\) bounds the local logarithmic change, the old conditional density is at least \(e^{-S_x}\) times the new one. Keeping the new sample with probability \(e^{-S_x}\) and otherwise sampling the residual law proves the assertion, since \(1-e^{-S_x}\leq\min\{1,S_x\}\). Separation of the integration squares permits independent decisions.

For a check of inner radius \(r\), start with \(W=S_{2ar}(z,\alpha)\setminus S_{r/2}(z,\alpha)\) and use \(W_k=W\ominus M_1s_k\), where erosion removes a layer along both boundary components. Choose \(M_1s_{k+1}\geq M_1s_k+C_1s_{k+1}\) and then \(M_0\) large enough that \(r\geq M_0s_k\) leaves the smaller proportional collars fixed above. Raise the first retained ring radius by a fixed factor if necessary so all enlarged stencils fit in disjoint collars.

For marks \(M\) from just one reversal and a spin sample \(\sigma\), let \(\mathcal C(\sigma,M)\) be the configurations agreeing with \(\sigma\) outside their marked squares. Define \[\begin{aligned} T^+(\sigma,M)&=\mathbf 1\{T=1\text{ for some configuration in } \mathcal C(\sigma,M)\},\\ D_T(\sigma,M)&=\mathbf 1\{T\text{ is nonconstant on }\mathcal C(\sigma,M)\}. \end{aligned}\] With \(\pi_{k-1,\eta}\) the mark law for reversing step \(k-1\), set \[ B_{k,T}(\eta)=\sup_\zeta\mu_{k,\eta,W_k}^{\zeta}(T),\qquad B^+_{k,T}(\eta)=\sup_\zeta (\mu_{k,\eta,W_k}^{\zeta}\otimes\pi_{k-1,\eta})(T^+),\quad k\geq1. \tag{49}\] Only \(B_{0,T}\) is used at level zero. Each supremum is a finite maximum: only finitely many exterior variables can enter the chart, and their alphabets are finite. It is consequently measurable and depends only on primitive environmental variables in the chart and a fixed \(O(s_k)\) neighborhood. Taking the supremum before environmental expectation preserves this locality and bounds arbitrary mixtures of exterior assignments: each fixed exterior kernel uses the same local primitive variables, so maximizing over exterior values introduces no additional environmental variables.

Closing the fixed-shape bootstrap

Lemma 7 now applies to every prescribed finite family of buffered polygonal tests. Such a family changes its constants \(C_T,a_T\), but does not change \(x_{\rm cr}\). The remaining convergence arguments use the elementary consequence of geometric spacing \[ \sum_{j<K}X_j\min\{1,(s_{j+1}/r)^c\}+X_K\longrightarrow0 \tag{56}\] as \(r,K\to\infty\), provided \(s_K/r\) stays between fixed positive bounds. Indeed the sum of the weights is uniformly bounded. For \(j\geq m\) bound \(X_j\) by \(\sup_{j\geq m}X_j\), and for the finitely many \(j<m\) let \(r\to\infty\). Then let \(m\to\infty\). This is where decay, rather than summability, is sufficient.

The three geometric consequences

The initializations and their scale outputs satisfy every array hypothesis used here. Proposition 4 makes the uniform bound on \(X_k\) in (43) as small as needed by reducing \(\varepsilon_0(\rho)\), and makes \(X_k\) decay. The fixed threshold \(x_{\rm cr}\) depends only on the three check shapes and the scale parameters. Thus this reduction of \(\varepsilon_0(\rho)\) precedes the choice of a final domain, test family, or approximation sequence. Proposition 6 places these conclusions on both lattices at \(\beta_*\). The local comparison will give FK annular barriers; the torus comparison and four-passage exclusion will control the interface.

Annular barriers at the matched temperature

Dual arms above the matched temperature

The local comparison supplies no rate in (61). We first obtain a useful rate by using disjoint bond sets in many separated annuli at each center. The resulting environmental exceptional probability decays faster than every power of the box size. Outside this exceptional set a layer exploration has small revealment, and the weighted derivative strengthens polynomial arm decay to a stretched exponential bound.

Percolation and plus magnetization

Boundary conventions and signed geometry

Fix a bounded Jordan domain \(D\), distinct marks \(a,b\in\partial D\), and the deterministic approximations in Theorem 1. Write \(D_n\) for their domains in physical coordinates, after multiplication by \(\delta_n\). A lattice Jordan domain here is a full square-lattice domain: bonds have not been deleted in its interior. One standard convention surrounds the union of closed square cells centered at the free spin sites, fixes the exterior nearest neighbors, and retains every bond incident to a free spin. The interface then has a local half-edge at each boundary change. Equivalently one can use a lattice polygon made of plaquettes, with the boundary vertices pinned and all interior vertices free. The argument below applies to both conventions. In the first convention the exterior pin assignment is an actual assignment to sites, so repeated incidences at one site must have the same value. The marked boundary changes and any half-edge endpoints differ by at most \(O(\delta_n)\).

We may take the boundary homeomorphisms to match the marks exactly. Indeed, their inverse images of the discrete marks converge to \(a,b\); in a fixed circle parametrization, precompose by circle homeomorphisms converging uniformly to the identity and taking \(a,b\) to those inverse images. After passage to a subsequence the designation of the plus arc is fixed. Denote the open limiting arcs by \(I_+\) and \(I_-\). Thus boundary order and signs, as well as positions, converge. Winding number stability then implies that each compact subset of \(D\) is eventually inside \(D_n\), and each compact subset of \(\mathbb C\setminus\overline D\) is eventually outside \(D_n\).

Let \(\Gamma_n\) be the target contour, oriented from the mark near \(a\) to the mark near \(b\). Its conditional law given the original environment \(\omega\) is \(Q_{n,\omega}\). We use the prescribed NE/SW resolution, or a simple drawing of the same resolution inside small vertex disks. Matching the contour pieces in their traversal order changes \(d_{\rm curv}\) by \(O(\delta_n)\). The resolved graph has degree two except at its two marked ends, so this drawing of its connecting component is a simple arc. Spin paths use the fixed triangulation by NW/SE diagonals from the crossing section. In the bulk its zero contours have precisely these resolved connections.

We next describe the boundary placements used in [22] and [23]. To obtain the lower pure comparison, move the middle plus boundary outward and the minus boundary inward. Two cuts allow comparison through a torus: a plus crossing screens the comparison domain’s plus boundary, and a minus crossing screens the target’s minus boundary. After revealing these crossings, the remaining boundary incidences have the orders needed for two comparisons with identical couplings. The strips on the central side of both cuts will transfer signed barriers between the domains.

Fix a plane Schoenflies homeomorphism \(\Psi\) taking the unit disk to \(D\), \(1,-1\) to \(a,b\), and the upper semicircle to \(I_+\). These coordinates describe topology only; all tests below are polygonal tests in the physical plane.

For small \(h>0\), first take a radial boundary equal to \(1+3h\) on \([4h,\pi-4h]\), equal to \(1-3h\) off \((3h,\pi-3h)\), and affine on the two intervening angular intervals. Its image under \(\Psi\) encloses a preliminary domain favoring minus. Put its marks at angles \(6h,\pi-6h\), and give the intervening upper arc the plus sign. The remaining arc is minus. A minus cutting corridor \(C_h^-\) follows \(\rho=1-h\) on \([\pi-h,2\pi+h]\), bending outward at its ends past \(\rho=1+h\); it is outside this comparison domain. A plus corridor \(C_h^+\) follows \(\rho=1+h\) from \(5h\) to \(\pi-5h\), bending outward at its ends past \(1+4h\). It is outside \(D\), and its caps cross the minus shoulders of the comparison boundary. Give the corridors small positive widths and slightly larger, mutually disjoint reveal regions on their exterior sides. Their caps extend beyond the boundaries they cut. The minus reveal region stays outside the comparison domain; the plus reveal region stays outside \(\overline D\). The central side of a cut means its side toward \(\Psi(0)\).

Replace this finite configuration by sufficiently close polygonal arcs in the physical plane, preserving order, the cap crossings, and all the stated positive clearances. Such approximations can be made inside the transported rectangular neighborhoods of the arcs; this is the deterministic construction in the cited section of [23]. Call the polygonal domain \(A_h^-\) and its marks \(a_h^-,b_h^-\). Choose the approximation errors to vanish with \(h\). Its boundary parametrizations therefore converge uniformly to those of \(D\). Rotating the coordinate construction by \(\pi\) and interchanging signs gives a domain \(A_h^+\) favoring plus, still oriented from the mark near \(a\) to that near \(b\).

An allowed plus strip is a polygonal topological rectangle whose closure lies inside \(A_h^-\), on the central side of both cuts, and is disjoint from the closed reveal regions. Its two end gates lie outside \(\overline D\) on the enlarged plus side. Its closure stays in \(\Psi(\{\rho<1+h/2\})\) with a positive margin. All its contacts with \(\partial D\) lie on \(I_+\), away from the marks. In a target sample a plus crossing of the strip is tested by filling its portions outside \(D_n\) virtually with plus. This specifies a geometric event; it changes no actual spin or Gibbs weight. Minus strips in \(A_h^+\) are defined with the signs reversed. Figure 2 distinguishes the cutting corridors from the strips they will protect.

At each fixed \(h\) these placements are valid for every sufficiently large \(n\). Boundary winding controls the compact clearances, and the marked boundary order controls signs near each contact. The caps prevent paths from going around the ends of a cut. This remains true for a free path ending next to an exterior pin: append its final incident bond before applying separation. With pinned boundary vertices that last bond already lies in the polygon. The difference between the two conventions is contained in one lattice layer and cannot defeat a fixed-clearance cap.

Conditioning at a fixed environment

For laws of vectors in \(\{0,1\}^m\), write \(\preceq\) for stochastic order with respect to coordinatewise order. We record the elementary coupling fact from [22], including its proof because its conditional marginal is essential here.

The environment has not been conditioned on \(e=1\). We condition spin laws separately at each fixed environment, including when deciding which environments require an arbitrary coupling.

From strips to ordered traces

To use a virtually filled crossing as a barrier, one needs exact noncrossing, rather than just two drawings at distance \(O(\delta_n)\). We give the boundary check from [23]. For a fixed finite strip list, its boundary contacts lie on compact signed arcs separated from the marks. Near a plus contact every exterior nearest neighbor of a free site is already plus. Fill the other exterior vertices of incident faces with plus for drawing purposes. In the cell convention, a boundary dual vertex meets one, two adjacent, or three interior cells; two opposite cells alone would violate the Jordan condition. With one interior cell, the two nearest exterior neighbors already have the filling sign, so filling the opposite exterior corner changes no bond incident to that cell. With two adjacent or three interior cells, all exterior corners are already nearest neighbors. Triangular interpolation consequently preserves every target disagreement edge and every resolved local connection. With pinned boundary vertices the same assertion follows directly by interpolating the plaquettes with their given arc values.

Redraw the target in the faces meeting the strips and one adjacent layer using these zero segments. They glue at the same edge midpoints, preserve traversal order, and avoid all corresponding monochromatic paths. Plus and minus contact sets are separated, so the two fillings can be made simultaneously. The half-edge ends and neighborhoods of the marks are untouched. Denote the resulting drawing by \(\widetilde\Gamma_n\); then \[ d_{\rm curv}(\widetilde\Gamma_n,\Gamma_n)=O(\delta_n), \qquad \widetilde\Gamma_n\text{ is disjoint from every selected barrier}. \tag{72}\] This changes the drawing only, including in a domain with narrow fjords: no exterior spin or incident interaction is removed.

We use the following pure geometric input from [22], proved in [23]. Given \(\eta_j\downarrow0\), one can choose \(h_j\downarrow0\) and a deterministic finite list of allowed strips of each sign at displacement \(h_j\) such that, outside pure probability \(O(\eta_j)\), a comparison chord selects a strip whose end gates it crosses with strict margins. Its omitted terminal pieces have diameter at most \(\eta_j\). The strip has cross-sections of diameter at most \(\eta_j\) following a simple prototype within \(\eta_j\) in curve distance of the remaining subarc. For sufficiently fine ordinary lattice approximations the same assertion holds, with errors \(O(\eta_j)\), and the signed side of the lattice interface supplies a monochromatic crossing of that strip.

Here are the relevant hypotheses and parameter order behind this input. The pure Dobrushin convergence to \(\mathop{\mathrm{SLE}}_3\) is used only in a fixed polygonal domain, with the full curve convergence supplied by [5, 17]; see also the pure input statements in [22, 23]. Uniform convergence of Jordan boundary parametrizations gives uniform convergence on the closed disk of the normalized conformal maps, by Radó’s domain-continuity theorem as stated and explained in [23]. In particular the hypotheses give a common boundary local-connectedness modulus. Uniform convergence transfers nearby points on the varying boundaries to nearby points of the limiting boundary; uniform continuity of the limiting inverse and of the parametrizations then controls a joining boundary subarc. The boundary crosscut proof of the conformal extension theorem upgrades kernel convergence using this modulus. Thus boundary fjords do not require an extra regularity assumption. Mapping a single disk \(\mathop{\mathrm{SLE}}_3\) by those maps shows that the continuum laws in \(A_h^\pm\) converge in the oriented curve metric to \(\mathop{\mathrm{SLE}}_3(D;a,b)\) as \(h\downarrow0\). Simplicity and avoidance of the boundary away from the two endpoints imply that compact middle subarcs stay inside \(D\). In the enlarged domain, take the last and first hits of \(\rho=1+h/3\) around an interior point and extend slightly at their ends. These hits are on the enlarged signed side, outside \(D\); the extensions stay below \(1+h/2\). Polygonal tolerances are chosen smaller than the gaps between these radial levels. This gives the allowed gates, and the omitted terminal diameters tend to zero. Thin rectangular neighborhoods of these simple subarcs can be chosen polygonal with strict traversal margins. Rational choices form a countable open cover of the favorable curves; a finite subcover in probability loses at most another \(\eta_j\). To see the signed crossing in a fine lattice approximation, follow the selected interface traversal through its successive interpolation triangles. Their plus-side vertices are connected by plus edges of those triangles, with common vertices or edges across each consecutive face. They form a plus walk at distance \(O(\delta_n)\) from the traversal. The positive long-side clearance keeps this walk inside the strip; the transverse crossing margins carry it beyond both end gates. Crop it at those gates and erase loops. Backtracking of the interface does not affect this existence argument. The entire finite list is fixed before Proposition 16 is applied. The sampled curve chooses its strip only afterwards.

Choose increasing deterministic integers \(n_j\ge j\) so that for every \(n\ge n_j\) all fixed-\(j\) placements, pure approximation bounds, and both strip transfers have errors at most \(O(\eta_j)\), and the pure comparison laws are within \(\eta_j\) of their continuum laws in bounded-Lipschitz distance. Include the fixed-clearance exterior closures described below among these requirements. This absorbs the possibly small \(c_{h_j}\) in the conditional coupling estimate. Set \[ j(n)=\max\{j\le n:n_j\le n\}. \tag{73}\] Before \(n_1\) make any choice. Then \(j(n)\to\infty\). Let \(L_n\) be the pure comparison chord in \(A_{h_{j(n)}}^-\) and \(U_n\) the pure comparison chord in \(A_{h_{j(n)}}^+\), both oriented from the mark near \(a\) to that near \(b\). Glue their couplings over the common pair consisting of the original environment and target spin sample, by sampling the two conditional comparison kernels given that pair. Each comparison marginal conditional on the original environment remains its deterministic pure law. Both marginal laws converge to \(\mathop{\mathrm{SLE}}_3(D;a,b)\), and with probability tending to one the target has both selected barriers.

All target traces eventually lie in a fixed compact planar disk. The space of its nonempty compact subsets is compact in Hausdorff distance. Together with tightness of \(L_n,U_n\) in the curve metric, this gives, along any further subsequence, a joint subsequential limit \[(\operatorname{tr}\Gamma_n,L_n,U_n)\Longrightarrow(K,L,U).\] Here \(K\) is connected, contains \(a,b\), and lies in \(\overline D\); these facts follow respectively from Hausdorff convergence, endpoint convergence, and the boundary winding argument. Both \(L,U\) are simple \(\mathop{\mathrm{SLE}}_3\) chords with interiors in \(D\). For such a chord \(C\), let \(P(C),M(C)\) denote the open components of \(D\setminus C\) adjacent to \(I_+,I_-\), respectively.

The barriers imply \[ K\subset\overline{M(L)}\cap\overline{P(U)}, \tag{74}\] where closures are in \(\overline D\). We spell out why this includes the open boundary arcs, following [23]. Represent the subsequential convergence almost surely and, after a further subsequence, make barrier failures summable. Erase loops in a successful plus crossing, join its ends to the gate centers within the exterior gate neighborhoods, and close it by radial paths in \(\Psi\) coordinates to a fixed circle \(\rho=\rho_*>1\), followed by its plus-side circular arc. The added paths have positive clearance from \(\overline D\) at each fixed \(j\); the thresholds \(n_j\) ensure that they are outside \(D_n\). Both target marks are on the zero-winding side: their outward escape paths lie beyond the ends of the extended plus arc. Exact disjointness in (72) keeps the entire target on that side.

The closed path need not be simple; its winding number is constant along any curve disjoint from it. Every crossing between the strip gates is homotopic within the rectangular strip, relative to its gates, to its prototype. Thus backtracking of that crossing cannot alter its winding about points outside the strip. Vanishing strip widths, prototype errors, and terminal diameters make this winding equal eventually to the nonzero winding of \(L\) closed on the plus side, on each compact subset of \(P(L)\). The fixed outer circular arc remains a positive distance beyond each compact subarc of \(I_+\), and the connectors tend to the two endpoint connectors. The same nonzero winding therefore holds in a neighborhood of every point of \(I_+\) other than \(a,b\). The target avoids all these neighborhoods. Its limit lies in \(\overline{M(L)}\), and the minus barrier gives the other inclusion in (74).

Consequently every joint trace limit satisfies \(K=L=U\). It remains to rule out repeated forward and backward traversal near this trace.

Passages control traversal order

The passage criterion below is proved in [23]; we include its proof to make the passage from compact sets to oriented curves explicit. It uses the interior estimate of Corollary 11. This is a concrete version of the role of multiple-crossing bounds in curve compactness arguments [1, 17].

Completion of the main theorem