Quenched SLE₃ limits for general weak random-bond Ising models

1 Introduction

The critical planar Ising model has a conformally invariant interface limit. For a random ferromagnet, the corresponding question asks whether weak microscopic disorder disappears from the macroscopic law when the environment is held fixed. In two dimensions the disorder is marginal, so a scale-by-scale decay estimate is needed in place of a uniform contraction. We extend the weak symmetric two-valued result of [22] to general bounded iid bond laws. The law need not be preserved by planar duality, so the proof must identify the critical temperature without assuming distributional self-duality.

1.1 Model and statement

Let \rho be a probability measure supported on [-1,1] such that

On the unoriented nearest-neighbor edges of \Z^2, let (\xi_e)_e be independent with law \rho, and set

Write \omega=(\xi_e)_e for the environment and \Prob_\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

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\Z^2 and define

The plus-state limit exists by ferromagnetic monotonicity. Throughout the paper, averaging a quenched law means integrating it against \Prob_\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\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

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

where \phi,\psi range over increasing homeomorphisms of [0,1]; curves at zero distance are identified. For probability laws on this space, set

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 \SLE_3(D;a,b).

1.2 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, Theorem 1 and Equation (1)]. 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, Theorems 1.1–1.2]. 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, Theorem 4.4 and Conjecture 4.5]. 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, Theorem 1.6]. Avérous and Mahfouf established high-probability crossing estimates for near-critical random-bond FK models, also with vanishing disorder for 1<q\le4 [2, Theorems 1.5–1.6]. 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, Section 8] 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, Theorems 1'–2']. 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, Equations (4)–(5)]. 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, Theorem 1.1], 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, Section 9].

1.3 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 (1.3). 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 (1.5).

2 Scale estimates for general local arrays

We first state the scale estimates that will be used to choose the temperature. These are results about local random potentials, rather than about a particular one-edge distribution. We use the exact energy charge, local transformation, moment estimates, and variance coordinate of [22, Sections 2–5]. The hypotheses below explain why those results apply to the present law \rho, although the main theorem of [22] concerns symmetric two-valued disorder. The pure-model inputs behind that construction are developed in [23, Sections 2–3].

2.1 The reference field and its energy charge

Let \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

The theorem “Exact pure energy charge” in [22, Section 2] provides the linear functional

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

This normalization permits charges at successive scales to be compared without accumulating a normalization error.

2.2 Arrays and the exact scale transformation

At an integer spacing s, a local array is a family f=(f_i)_{i\in\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: \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, \E\|f_0\|_\infty^{40}<\infty.

Here and throughout the scale argument, \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 \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, Section 3] 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

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

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.

2.3 Moment and variance coordinates

Write q_s=Q/s, and define

All covariances here are environmental. The covariance sum is finite by the dependence hypothesis. It is nonnegative because, for growing squares \Lambda\subset\Z^2,

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, Section 5, Definition “Variance coordinate”] begins with a finite random insertion F=\sum_{i\in I}t_i f_i, where I\subset\Z^2 is finite and the t_i are real parameters. Its normalized charge is

We take Taylor coefficients at zero for each fixed environment; the expansions below use only finite moments, not environmental exponential integrability. Since \E_0F=0 and q_s kills constants, the first three homogeneous terms are

The terms of degrees two, three, and four in the formal environmental variance of T(F) are therefore

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

The derivatives mean coefficient extraction from the displayed variance polynomial. For H>R_0, define

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, Sections 4–5], 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 (2.7) 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 (2.10) to keep the entire iteration small.

3 Two temperature tunings

The scale estimates select a temperature at which the perturbation decays. For a general law \rho, there is no reason for that selection to be self-dual. We therefore apply the selection separately to the physical family and to its exact dual, using the same physical inverse temperature as the parameter. This section constructs the two trajectories; their temperatures will be compared afterward.

3.1 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

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

where e^* crosses e. We identify the dual square lattice with a translate of the square lattice.

Use the scale-step parameters of Proposition 2.1 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, Section 7]. Both lattices use the spacings s_k=s_0L^k. The two input families, parametrized by a small number h, are

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

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 (2.3) give a deterministic bound on each potential at every fixed number of steps, uniformly on a compact interval of h.

Recall \sigma_\rho^2=\E_\rho\xi^2>0, and put

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.

3.2 Keeping the expanding coordinate between two walls

We now adapt the shooting argument of [22, Section 6] 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.

4 Matching the primal and dual temperatures

Proposition 3.2 supplies a small trajectory for the primal couplings at \beta_{\rm p} and another for the dual couplings at \beta_{\rm d}. We now prove that these physical temperatures coincide. The comparison takes place in the same bond environment on a torus. Exact duality relates the four spin partition functions, while FK monotonicity separates two distinct temperatures. The argument extends the torus identification in [22, Section 8, Proposition “Identification of the tuned parameter”]; it uses the two trajectories in place of symmetry of the disorder law.

4.1 FK differentiation at fixed environment

For a finite graph G=(V,E) and a partition \pi of its boundary vertices, let

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 (3.1). 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 (4.1) gives

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

Here C_{\rm DT} is an absolute constant; an independent randomized choice of tree is allowed. We use the monotonic-measure extension [9, Theorem 1.1] of the OSSS inequality [21]. Both (4.2) and (4.3) are applied at fixed environment, before any environmental expectation.

4.2 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=(\Z/N\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

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.

5 From decaying potentials to crossing comparisons

Propositions 3.2 and 4.2 give potential arrays with small, decaying moment bounds on both lattices at the same physical temperature \beta_*. We now turn this information into three geometric consequences: local conditional lower bounds, comparison of finite test vectors on tori, and exclusion of interior four-passage bottlenecks. The distinction between smallness and decay matters. Smallness supplies uniform passage estimates; decay makes the comparison errors vanish, although \sum_k X_k need not be finite.

The crossing propositions of [22, Section 7] are stated along its binary tuned trajectory. Their proofs use the local scale construction and independent primitive environment variables, together with

We give the extension to the arrays used here, including the conditional product estimate on which it depends. Throughout, the arrays form an actual trajectory of the exact local map of [22, Section 3], with its conditional implementations and reversals specified below. On a torus we use the same local formulas and independent primitive variables, so that every embedded stencil has its plane law. Merely assuming the displayed bounds for unrelated arrays would not suffice. Throughout the section s_k=s_0L^k; \eta denotes the enlarged environment, including the independent priorities and thresholds of the scale construction. The Gibbs field contains spins and a locally finite family of finite-valued auxiliary variables, whose reference law is a product. These auxiliary variables are distinct from \eta; their decorated Gibbs law need not be a product.

5.1 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, Section 2, “Iterated four-passage bound”, “Pure buffered RSW consequences”, and “Two collars around an annulus”; Section 5, “Two bulk passages confined to a half-plane”]. For some c_0>0, they give

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

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, Section 5, “Passages forced by a pivotal square”], 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

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.

5.2 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

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, Section 3]. 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

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

With \pi_{k-1,\eta} the mark law for reversing step k-1, set

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.

5.3 Closing the fixed-shape bootstrap

Lemma 5.1 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

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.

5.4 The three geometric consequences

The initializations and their scale outputs satisfy every array hypothesis used here. Proposition 3.2 makes the uniform bound on X_k in (5.1) 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 4.2 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.

6 Identification of the physical critical point

The matched temperature \beta_* was selected by the scale map. To identify it with the magnetization threshold, we first construct local FK barriers at \beta_* on both lattices. These barriers rule out percolation at the matched temperature. Above \beta_*, the dual odds decrease; a decision-tree argument then makes dual connections rare enough to force primal percolation. This is the local sharpness argument of [22, Section 8], applied to the exact dual family rather than to a distributionally self-dual model.

Write B_r(x)=\{y:|y-x|_\infty\le r\} on the lattice under consideration; radii may be rounded to integers. Let \phi_{\beta,\omega}^{\mathrm w} be the primal infinite-volume wired FK measure. It is obtained by decreasing limits of finite wired-box laws on local increasing events, and the limit does not depend on the exhaustion. The analogous notation with a star refers to the dual odds at the same physical parameter \beta.

6.1 Annular barriers at the matched temperature

6.2 Dual arms above the matched temperature

The local comparison supplies no rate in (6.2). 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.

6.3 Percolation and plus magnetization

7 Signed barriers and the quenched interface limit

By Proposition 6.3, the common tuning parameter is \beta_* =\beta_c^\rho(\varepsilon). We now use the finite-vector comparison of Proposition 5.4 to place the target interface between two pure comparison interfaces. The comparisons will preserve the pure marginal conditional on the original bond environment. This is the property that will yield convergence of quenched laws, rather than only convergence after averaging over bonds. Throughout this section, probabilities without an environmental subscript include spins and coupling randomness as well as bonds.

7.1 Boundary conventions and signed geometry

Fix a bounded Jordan domain D, distinct marks a,b\in\partial D, and the deterministic approximations in Theorem 1.1. 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 \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, Section 9, “Displaced domains and finite boundary barriers”] and [23, Section 7, “Fixed displaced domains and their boundary incidences”]. 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.

7.2 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, Section 9, Lemma “Conditioning and gluing at fixed environment”], 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.

7.3 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, Section 7, Lemma “Local drawing beside a signed strip”]. 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

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, Section 9, Lemma “Finite strips and limiting barriers”], proved in [23, Section 7, “From fixed strip tests to limiting barriers”]. 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 \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, Section 7, beginning of “From fixed strip tests to limiting barriers”]. 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 \SLE_3 by those maps shows that the continuum laws in A_h^\pm converge in the oriented curve metric to \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 7.2 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

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 \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

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 \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

where closures are in \overline D. We spell out why this includes the open boundary arcs, following [23, Section 7, “Forbidden sides, order, and identification of the trace”]. 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 (7.2) 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 (7.4).

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

7.4 Passages control traversal order

The passage criterion below is proved in [23, Section 7, Lemma “Progress coordinates and the full curve metric”]; we include its proof to make the passage from compact sets to oriented curves explicit. It uses the interior estimate of Corollary 5.5. This is a concrete version of the role of multiple-crossing bounds in curve compactness arguments [1, 17].

7.5 Completion of the main theorem