Quenched SLE₃ Universality for the Weak Random-Bond Ising Model

Proof. An even finite dual edge set determines a unique spin configuration equal to plus far away: the sign at a primal vertex is the parity of crossings of any path from that vertex to infinity. Evenness makes this parity independent of the path. Restricting the contour to E_V gives a bijection between the compatible spin configurations on V and \mathcal P_V. Across a primal bond, the spin product is -1 exactly when its crossing dual edge belongs to the contour. Thus

The first factor is constant over P. Every exterior-only energy is constant as well. Normalization therefore gives (1.5), whose formula does not involve P_o.□

Proof. For the invariant subspace use the displayed upper-limit estimates in [22, Section 3, “Compatible lattice coefficients and arbitrary boundary data”], dividing the companion’s density-normalized operator by d^2. The measured coefficient is

where W is the fixed exterior annular test there and g\ne0 is its radial degree-one coefficient. The remainder in the transfer norm is C d^{-3}+C/(dH^2), and that in coefficient compatibility is C/(dH^2). Choose H large enough for all the prescribed ratios, then the replacement accuracies, and finally s_* to absorb the uniform upper-limit errors into C d^{-3}. Center the companion’s fixed interior reference test under \E_0 and divide it by its measured coefficient to obtain the centered bounded right inverse. Centering changes neither that coefficient nor the transfer estimates. The radial spectral gap requires only quarter turns, so reflection averaging in this application is unnecessary.

For completeness, the same method proves (2.4). Before the angular averaging in the proof of the companion’s radial gap, the polynomial expansion has the form

Both displayed coefficients are rotation scalars. Before smearing, the second factor is c(\prod_i\sigma(z_i))=S(z_1,\ldots,z_{2m})A_2(z)/A_0(z), where

All distances in this expression are rotation invariant. Thus both vanish on the even zero-quarter-turn-average subspace. A polynomial self-pairing there is the integral of r^a against a positive spectral measure and is O_F(r^2). Positive mass in any compact subinterval of [0,2) would contradict this bound as r\downarrow0. Polynomial density therefore places the whole spectrum of this subspace in [2,\infty), giving the uniform radial bound r^2\|F\|_2\|G\|_2.

Apply bounded replacement to the observable and to the buffered boundary density used in the proof of uniform transfer. Project the first proxy with I-\mathcal A; this changes its norm and the replacement error by at most a factor two. The radial bound just proved then gives C d^{-2}, uniformly over the boundary assignment. The same finite-list choice of accuracies and threshold makes the lattice inequality uniform in s and in the observable. This proves the lemma without a spatial symmetry assumption on the original even observable.□

Proof. Start with a bounded right inverse v_0 of \ell_{s_*} and propagate it by

Equations (2.2)–(2.3) show by induction that \|v_j\|_\infty stays bounded and m_j=d^{-1}(1+O(d^{-2})). Indeed, choose a fixed bound larger than four times the constant in (2.2), then choose d so large that the d^{-2} errors preserve it.

Write f_j=a_jv_j+k_j, where \ell_{s_j}(k_j)=0. Transfer has the triangular form

The last estimate follows by subtracting v_{j+1}\ell_{s_{j+1}}(P_{s_{j+1}}k_j) from the transferred kernel vector. Put M_{j,n}=\prod_{h=j}^{n-1}m_h, with M_{j,j}=1, and define

For n=j the product of K’s is the identity. The series converges in operator norm because the ratio of its successive bounds is O(d^{-2}); its correction to a has norm O(d^{-2})\|k\|_\infty. Shifting the series proves exact compatibility with multiplier m_j.

If \beta_j(f)=0, the scalar at each later time is the negative of the remaining tail and has magnitude at most C d^{-2}\|k_n\|_\infty. Hence the whole vector contracts at rate (C d^{-3})^{n-j}. Enlarge d so that this is bounded by d^{-w(n-j)}. If \beta_j(f)\ne0, division by M_{j,n} shows that its scalar converges to the nonzero value \beta_j(f) while its kernel part tends to zero. It therefore enters (2.6). Finally, for a vector in that cone,

Choose M first, larger than the limiting bound, and then d large. This proves invariance of the cone.□

Proof. If a transferred B belonged to the stable hyperplane, then (2.5) and the Markov property, tested against a distant B_x, would give \E_0BB_x=O(|x|^{-w}). This contradicts (2.1), since w>2. The cone assertion follows from Lemma 2.2.

We next show that, for one sufficiently large fixed T, any centered unit-norm cone observable f_s satisfies

uniformly for all sufficiently large s; choose T divisible by 8, and \tau_x translates an observable by x. The value c_T can be chosen of order T^{-2}.

To see this, choose nonzero nonnegative smooth bumps \psi_1,\psi_2 in two disjoint regions strictly inside the unit square. Let Y_s be the quarter-turn average of \Phi_s(\psi_1)\Phi_s(\psi_2), centered under \E_0. The corresponding continuum spin-smear product, averaged under quarter turns and centered, has nonzero scalar coefficient. For two unsmeared pairs z_1,z_2 and Te_1+y_1,Te_1+y_2, the squared four-spin formula of the companion gives

Here the neutral assignments within each pair give the product term; their cross logarithms are O(T^{-2}) and cancel to first order after sign summation. The two assignments with charge 2 and -2 on the respective pairs give the displayed strictly positive coefficient after taking the positive square root. The expansion is uniform on the separated bump supports, so it survives smearing and averaging.

Take a sufficiently large fixed A and put V_s=\chi_A(Y_s)-\E_0\chi_A(Y_s), where \chi_A(t)=\max\{-A,\min\{A,t\}\}. The nonzero scalar coefficient survives for all sufficiently large A: the moment bounds give ordinary L^2 convergence of the truncated continuum test, and the radial coefficient is continuous in that norm. Its measured lattice coefficient is then bounded away from zero once H and the lattice threshold are large.

To see why A may be fixed before T, put \varepsilon_A=\sup_s\E_0|Y_s-V_s|, which tends to zero by the moment bounds. Buffered comparison gives

The last constant is independent of A, since \E_0|V_s|\leq2\E_0|Y_s|. Apply the overall C/T transfer bound of Lemma 2.1 to both factors, now conditioned on these radius-2s collars, and integrate in the two disjoint radius-Ts/4 squares. Expanding the difference of the two products yields

Thus a fixed large A preserves the positive T^{-2} coefficient. For all large fixed T, the covariance of V_s is at least cT^{-2} once the lattice threshold is large enough.

Now put a_s=\ell_s(f_s)/\ell_s(V_s). The cone condition and the uniform coefficient bounds give 0<c\leq|a_s|\leq C. The observable f_s-a_sV_s has measured coefficient zero. Transfer it and the other factor into the disjoint squares of radius Ts/4 about their centers. Lemma 2.1 bounds the first factor by CT^{-3} and the other centered factor by CT^{-1}. Thus

which proves (2.8). In this argument choose the truncation, then T, then a common measurement distance for the finite list d,T/4,T/8, and finally the lattice threshold. The ratio T/8 is used after transferring a truncation error to its radius-2s collar. Reapply Lemma 2.2 with these choices; its constants are uniform and its conclusions are unchanged. For this final choice, the covariance contradiction at the start of the proof again shows that P_sB is not stable, so it eventually enters the newly constructed cone.

Apply (2.8) to f_s=P_sB/\|P_sB\|_\infty. By simultaneous Markov integration in the two disjoint squares,

The cone lower bound and (2.1) give the upper bound in (2.7). The inequality |\E_0[f_s\tau_{Ts e_1}f_s]|\leq1 gives its lower bound. We first obtain these estimates along the geometric scales. For an intermediate radius, the tower property bounds its norm between the norms at the two adjacent scales, proving (2.7) for every large s.□

Proof. We first work at the origin on the geometric scales and with quarter-turn-invariant centered functions. By Lemma 2.3, the vector E_j=P_{s_j}B lies eventually in the cone and has norm comparable to s_j^{-1}. Since \beta_j=\ell_{s_j}+O(d^{-2}) and d is large,

Define temporarily \widetilde Q(f)=\beta_j(f)/\beta_j(E_j) for a function supported in B_{s_j}. Exact compatibility of both numerator and denominator makes this definition invariant under transfer to later geometric scales. This is also consistent when the same function is regarded as belonging to a later support space: the difference between it and its transfer vanishes after the next transfer, hence has zero \beta by compatibility. Its kernel is exactly the stable hyperplane, and |\widetilde Q(f)|\leq Cs_j\|f\|_\infty.

Increase the fixed starting scale once so that the calibration holds on the whole grid. For a function supported in radius s above this threshold, use the next containing geometric scale r, so that s\leq r\leq ds. At that scale compare f with \widetilde Q(f)P_rB, whose norm is at most C\|f\|_\infty by (2.7). Their difference is stable. Lemma 2.2, followed by norm contraction between adjacent scales, proves (2.12) with v=w and \widetilde Q in place of Q. Radii R within a fixed factor of s are absorbed by increasing the constant.

For a general even function, apply this construction to its quarter-turn average and set \widetilde Q(f)=\widetilde Q(\mathcal A_s f). The complementary component contracts at rate (Cd^{-2})^j by (2.4). Choose d large enough that this is at most d^{-uj}. This proves the general remainder estimate with v=u. Together with \|P_RB\|_\infty\leq C/R, it proves (2.11).

For fixed f, pair its remainder with B_x and integrate both factors in disjoint squares of radii proportional to |x|. The remainder has norm O_f(|x|^{-u}), and the transferred B_x has norm O(|x|^{-1}). Since u>1, their covariance is o(|x|^{-2}). Equation (2.1) therefore gives

This proves existence of (2.9) and identifies Q=\widetilde Q. In particular the definition is independent of the geometric grid and its starting scale. Translation and square invariance follow from this intrinsic limit and the corresponding symmetries of \mu_0; the limit is unchanged by a fixed displacement of its distant test. Equation (2.1) gives Q(b_e)=1 for every edge. Finally the Markov property preserves the covariance with every sufficiently distant B_x, so it preserves Q under transfer through any containing square. This also establishes the stated results about arbitrary centers. For the finitely many radii below the fixed threshold, transfer first to that threshold and enlarge the universal constants. Integrating independent auxiliary variables at the outset proves the asserted extension to them.□

Proof. Put r=|x-y| and integrate in disjoint squares of radius a small fixed fraction of r. Write F,G for the transferred observables and U_x,U_y for the corresponding transfers of B_x,B_y. Theorem 2.4 gives

where v=u, or v=w in the symmetric case. Notice that these estimates compare with the bond observable after transfer to a containing scale; the untransferred Q(f)B_x can have norm of order s. It follows that

Since v>1, (2.1) proves (2.13). Its error is uniform: r=ls\geq l, the energy remainder tends uniformly to zero outside growing disks, and |q_s(f)|+|q_s(g)|\leq C_D.

Simultaneous integration in the disjoint squares also preserves the charge of their product. Indeed it preserves its correlation with every sufficiently distant energy test, so (2.9) applies. All transferred products fit in a square of radius Cr. The charge bound, applied to the same product difference as in (2.15), therefore gives

It remains to prove

For distinct edges, Kramers–Wannier duality at the critical coupling gives

Here e_i^* is the crossing dual edge. To check the centering directly, differentiate the finite planar partition-function duality [18] in the distinct couplings. Its coupling-dependent prefactor is \prod_e(\sinh(2K_e))^{1/2}, and \sinh(2K_e)\sinh(2K_e^*)=1. At K_0 the logarithmic prefactor derivative is \sqrt2 and \dd K_e^*/\dd K_e=-1; thus \partial_{K_e}-1/\sqrt2 becomes -(\partial_{K_e^*}-1/\sqrt2). Distinct derivatives introduce no contact terms. Exhausting the plane and using uniqueness of its pure critical state proves (2.18).

Apply (2.18) to the three separated bond averages in the definition of Q(B_xB_y). After identifying the dual lattice with the original lattice, the distant dual average B'_z is centered and has charge one. Thus B'_z-B_z is centered and has charge zero. With x,y fixed, separately transfer the centered product B'_xB'_y-\E_0(B'_xB'_y) and this distant difference into squares of radii proportional to |z|. Their norms are respectively O_{x,y}(|z|^{-1}) and O(|z|^{-u}), by Theorem 2.4. Their covariance is therefore O_{x,y}(|z|^{-u-1})=o(|z|^{-2}). Replacing the distant dual average by B_z leaves the charge limit unchanged, and consequently

where the primed average consists of the four dual edges surrounding the original site.

There are four identifications obtained by the translations h=(\pm\frac12,\pm\frac12). They send the dual average about x to the boundary-edge averages of the four unit plaquettes incident to x. Let B'_{i,x}, 1\leq i\leq4, denote the resulting observables, and put

The observable C_x is quarter-turn invariant about x, has charge one, and is supported in a fixed-radius square. Each D_{i,x} has charge zero, and their average vanishes. Hence

The charge bound after separate transfers gives

For the first estimate both factors have zero charge and decay as r^{-u}. For the second, expand the difference with one factor C-B; it is quarter-turn invariant and has zero charge, while the other centered factor decays as r^{-1}. The common-support charge bound supplies the factor Cr in each case. Averaging the negative duality identity now gives 2|Q(B_xB_y)|\leq C(r^{-w}+r^{1-2u}). Since 2u-1>w, this is (2.17).

Finally, its contribution in (2.16) is at most

This proves both charge bounds uniformly for s\geq1.□

Proof. First integrate the auxiliary variables and put

The function W is positive and even. The fixed-ratio buffered comparison of [22, Section 2, Proposition “Buffered and small-patch comparisons”], applied to B_m and \partial B_{2m}, gives a constant K\ge1 such that

Indeed, the conditional law on B_m has pointwise density between K^{-1} and K relative to its full-plane marginal: the latter is a mixture of the same conditional laws. Integrating the positive function W proves the displayed bounds, regardless of its size.

Spin-flip symmetry gives h(-\xi)=h(\xi), and the tower property gives \E_0h=1. If t=\|G\|_\infty, then both W and Z lie between e^{-t} and e^t. Consequently

Apply the even transfer estimate in Theorem 2.4 to the centered even function h-1, supported in B_{2m}. Since R\ge8m, it gives

The Ising Markov property, followed by conditional expectation, yields

The logarithm is K-Lipschitz on [K^{-1},\infty). Its oscillation in the last display is therefore at most 2K\|P_R(h-1)\|_\infty, which proves (2.19).□

Proof. A parent whose F_i meets I_a is within 6 parent units of a. Suppose a comparison edge has endpoint neighborhoods meeting two selected squares, centered at a and b. An edge of length at most 12 cannot do this, since it would give \|a-b\|_\infty\le24<D. For a longer edge, an endpoint i within 6 of a selected parent a\ne i has already lost the shorter comparison \{i,a\}. Its martingale difference is therefore identically zero. For both endpoint contributions to survive, both endpoints would have to be selected; their common comparison coin makes this impossible. We may consequently declare the support of a normalizer using only those endpoint supports whose differences have not vanished. These declarations depend on coins and priorities, not on the functions being Taylor-expanded. Selected F_j themselves have radius less than one parent unit, so their separation is immediate.□

Proof. Since \E_eU_e=0, every coin normalizer starts at degree two in the child potentials. The derivative of (3.3) at zero is conditional expectation. These facts give (3.4). Averaging the transfer in the full-plane reference law gives zero by the centering of each child; averaging A_j also preserves this centering. On a small ball, real conditional exponentials are uniformly bounded above and below. The ordinary power series for the exponential and logarithm consequently give the asserted bounds in the supremum norm. There are only finitely many terms in each stencil, independently of s.□

Proof. Choose C large enough to contain the comparison, inactivity, and isolation ranges in parent units. Then choose L so large that old supports overhang a parent by less than 1/10 parent unit and 100d_0/L<1/10. Every selection or status check, local allocation, and conditional partition function then uses a fixed parent stencil. Old environment dependence adds only an overhang C/L in these units. The new priorities and thresholds are independent and located in the same stencil. Increasing C once, before fixing L, covers all these supports. No integration requires environment variables beyond the terms that touch its finite square.

For any probability kernel P and real bounded H, |\log Pe^H|\le\|H\|. The same inequality and its Lipschitz version bound (3.3) and (3.5) by sums of the input norms. The factors p^{-1} and the finite numbers of terms depend only on the fixed step. Centering at most doubles this bound, proving (3.7). Spin flip commutes with every operation. The symmetric choices and allocation rules give the stated invariance in distribution. Exactness follows from the normalized decoration, (3.3), and (3.6). Pure conditional kernels in an embedded square depend only on its boundary assignment, so the same identities hold on a torus. We always subtract the same full-plane centering constants, rather than torus means.□

Proof. Use the full local comparison stars. If a parent’s complete decoration and allocation stencil leaves W, set its F_j to zero in every U_e and retain its original child logs separately. Thus every fresh coin with a nontrivial likelihood is inside W and is sampled. Skip integrations whose stencils leave W. Include retained terms in any interior integration they meet. Skip special integrations near the boundary and keep active ones separated from them as above. Every operation affecting the eroded chart then uses precisely its full stencil. All remaining discrepancies are supported in the removed boundary layer. The argument applies also to annular charts.□

Proof. First sample the entire output Gibbs field, including the new coins. Independently sample a uniform variable at every candidate parent center, and mark j when it is at most p_j=1-e^{-C_LS_j}, with S_j the environmental norm sum in (3.8). These parameters depend only on the environment, so the marks have exactly the independence stated in the proposition, before any further conditioning.

Condition now on the new coins and the common exterior of the updated squares. The ratio of the old conditional density to the new one is bounded below by e^{-C_LS_j}, where S_j is the local norm sum in (3.8). This follows from the real log bounds used in Proposition 3.3; for a special square it applies to the tilt from B_A to B_A+G_A. If two laws satisfy \mu\ge(1-p)\nu and p>0, sample from \nu, retain that sample with probability 1-p, and otherwise sample from [\mu-(1-p)\nu]/p. The Bernoulli decision can be independent of the \nu sample; when p=0 the laws agree and the sample is retained. Use this residual resampling in each marked updated square. Conditional factorization over the separated squares proves that the resampled field has the old conditional law. Unused centers keep their dummy marks and cause no change. Each update lies inside its radius-5s' square. Finally discard the conditionally decorated coins. Their decoration was normalized at each old field, so this recovers the input marginal. Taking environmental expectation in (3.8) gives the last bound.□

Proof. Only a finite neighborhood of the insertion has nonzero decorations or replacement terms. Every observable outside a larger buffered region has the same expectation against the normalized input and output densities, because the decorations and replacements preserve its unnormalized pairing and the partition function. Apply this to a pure bond tending to infinity. Theorem 2.4 identifies the limiting pairing with Q, so the total lattice charge is unchanged. Dividing by the output scale Ls instead of s gives the factor L^{-1}. Scalar centering constants cancel in D_f. Analyticity near zero gives the coefficientwise assertion.□

Proof. Write P=2r and expand \|S\|^{2r}=(\sum_{i,j}\langle V_i,V_j\rangle)^r. For each tuple of 2r occurrences, connect two occurrences when their indices coincide or are neighbors in the dependency graph. A singleton component has expectation zero by centering and independence. If all components have size at least two and one has size greater than two, there are at most r-1 components. Bounded graph degree gives at most C_{P,\mathrm{dep}}N^{r-1} such tuples: choose one index per component, then the remaining indices along a spanning tree. Hölder’s inequality bounds each expectation in absolute value by a^{2r}.

For the remaining tuples every component is a pair. Independence factors their expectation into contractions of pair covariance operators. Sum these contractions over all pairings and allow the indices belonging to different pairs to be adjacent. The added tuples have merged components and incur the same C_{P,\mathrm{dep}}N^{r-1}a^{2r} error. The unrestricted sum is exactly the Wick expression for \E\|Z\|^{2r}, where Z is a centered Gaussian Hilbert vector with the covariance operator of S. This operator is positive even though individual cross-covariances need not be positive. Diagonalizing it gives \E\|Z\|^{2r}\le C_P(\E\|Z\|^2)^r =C_P(\E\|S\|^2)^r. Finite-dimensional projection and monotone convergence give the same conclusion in a general Hilbert space. Finally bounded dependence gives \E\|S\|^2\le C_{\mathrm{dep}}Na^2.□

Completion of Proposition 4.1. The last line follows by combining (4.12), (4.14), and (4.19):

Choose L large enough that the first two coefficients, and the coefficient CL^{2-w} in (4.11), leave a strict margin below one. This is possible because u>1, w>2, and P>2. Choose b next to put flag-free stencils in the Taylor neighborhood and make C_Lb smaller than the remaining margin. Finally choose x_* so that C_{L,b}X is smaller than that margin for X\le x_*. Increasing the resulting contraction factor to one common \rho<1 proves all three assertions.□

Proof. On the unflagged stencils, the fifth-order Taylor remainder and Hölder’s inequality give O_L(X^5). On flagged stencils use the raw bound for the actual outputs and (4.13). Products of four fourth-order Taylor polynomials have degree at most 16<P, so the same bound controls every term there. Terms of input degree between five and sixteen have expectation O_L(X^5) when X\le1. Products of separately taken expectations obey the identical estimates.□

Proof. The quadratic coefficient is C_2(i,j)=2\Cov(q_s(f_i),q_s(f_j)). Its contribution is therefore exactly W, because covariances vanish past the dependence range. Every remaining coefficient has three or four potential occurrences. Their supports fit in a ball of radius O(Hs). The charge bound in Theorem 2.4, the boundedness of pure expectation, and Hölder’s inequality bound each coefficient of degree d by C_H X^d. There are finitely many tuples with first slot zero and diameter at most H.□

Proof. Decompose f_i=\bar f_i+\widetilde f_i, where \bar f_i=\E f_i. An occurrence of \widetilde f_i has size order one, and an occurrence of \bar f_i has size order two. Products of total size order at least five contribute O_H(z^5) by Hölder’s inequality. For the remaining coefficients connect two occurrences when their cell indices are within R_0. A connected component containing at most four occurrences has bounded diameter. Moreover, a component consisting of a single centered occurrence has zero environmental expectation in every term of the covariance expansion. Thus, once H is large, a nonzero coefficient in the shell has precisely two distant groups: either a centered pair and one mean, or two centered pairs. Each has total size order four. This argument applies also to the products of means subtracted in a covariance.

Call the two groups A and B. Let 1+g_A and 1+g_B be their separately normalized insertion densities, expanded formally in their own indeterminates. Every coefficient of g_A,g_B is pure centered. Define

Multiplying the two densities and normalizing again gives the exact formal identity

If the anchor distance is l, Proposition 2.5 gives, coefficient by coefficient,

The estimates carry the product of the norms of the local coefficient functions. All group diameters are bounded, so their constants are uniform in the slots and in s.

The variance of T_A+T_B has no coefficient using both independent groups. Contributions with two cross factors h_{AB} or c_{AB} have shell sum bounded by C H^{2-2u}z^4=o_H(1)z^4. For a single h_{AB} we use the stronger, symmetric part of Proposition 2.5. To justify its applicability, first replace the cutoff on total diameter by the cutoff on anchor distance and sum all bounded relative positions inside each group. Environmental factorization turns the required coefficient into a finite sum of bilinear charge pairings of deterministic functions. More explicitly, the two possible local sums, besides \bar f_x, are

The pair-plus-mean sector gives pairings q_s(D_x\bar f_y); the pair-plus-pair sector gives q_s(D_xM_y), with their exchanges and the fixed scalar factors from Taylor expansion. This list follows by taking the linear charge from one group and the term linear in that group in h_{AB}; the other group supplies its mean or its quadratic normalized density. Terms leaving a centered singleton vanish as above. All displayed functions are pure centered, and D_x,M_x have norm O(z^2). Rotation of the offset sum preserves both its index set and the environment law, while charge is a rotation-invariant scalar. Thus each function is quarter-turn invariant about its anchor. The resulting bound is C l^{-w}z^4, whose shell sum is O(H^{2-w})z^4=o_H(1)z^4.

The replacement of diameter by anchor distance affects only strips of bounded thickness at the two ends of the shell. There are O(H) possible anchor displacements in these strips, and the unsymmetrized bound O(H^{-u}) gives an error O(H^{1-u})z^4. Thus the replacement and its reversal both have vanishing error as H\to\infty.

Only the single-c_{AB} contribution remains. Up to size order four it is

Its four occurrences must all be centered; a mean would increase the size order. Write U=T_{A,1} and V=T_{B,1}. These are independent centered random scalars. At bidegree (2,2),

The o(1) in (5.10) gives o_H(1)z^4 after shell summation, since L is fixed and \sum_{H<|y|\le LH}|y|^{-2} is bounded.

We finally spell out the slot count. Put a_{ij}=\Cov(q_s(f_i),q_s(f_j)). Suppose the four labeled slots split into the far pairs \{1,2\} and \{3,4\}. The coefficient of t_1t_2 in \E U^2 is 2a_{i_1i_2}, even when the two cell indices coincide; the occurrences still have distinct indeterminates. The other pair contributes 2a_{i_3i_4}. Hence the polarized normalization contribution is

There are three partitions of four labeled slots into two unordered pairs. Their sums agree by permutation of the dummy slots, and a tuple in a sufficiently distant shell has at most one such far pairing. The prefactor from (5.5) is 1/4!, so the numerical coefficient is -16\cdot3/4!=-2. For the pairing just displayed, anchoring i_1=0 and writing y=i_3 gives

Both sums are finite, and stationarity makes the second equal to W. Changes of the cutoff caused by bounded shifts of anchors have the strip bounds already proved. A change of anchor in the kernel itself also has vanishing error: for bounded \delta and |y|\asymp H, \bigl||y+\delta|^{-2}-|y|^{-2}\bigr|\le C|\delta|H^{-3}, whose shell sum is O(H^{-1}). This proves (5.7).□

Proof. First use the regular map and retain the child indeterminates. For finitely many labeled inputs its normalized total insertion charge obeys

by Corollary 3.6. One can also see the normalization directly: the conditional coin likelihood integrates to one at every fixed old configuration, and each conditional integration preserves pairing with a test outside the affected region. Pairing with a distant pure bond identifies Q by Theorem 2.4. Centering the allocated output logs changes only a scalar, which disappears from the normalized insertion. The factor L^{-1} in (5.14) is exactly the change from Q/s to Q/(Ls).

This argument holds for every fixed realization of the fresh priorities. Taking environmental variance and then Taylor coefficients therefore preserves complete coefficients, including their mean subtractions. The variance factor L^{-2} cancels the L^2 children per parent when sums are taken per unit area. If an allocation distinguishes positions within a parent, take the area average over complete parent periods. There is no assumption of invariance of individual allocated coefficients under child translations.

We must compare the two cutoffs. Every output Taylor coefficient uses children in a fixed stencil. Thus classifying a coefficient by parent diameter H rather than child diameter LH changes its classification only when the child diameter lies in a strip of width O_L(1) about LH. Keep together the full covariance coefficient for each fixed tuple of output arguments, including all its mean subtractions, and only then expand its arguments in child slots. The parent cutoff is constant on precisely this grouping. More explicitly, write R_j for the regular output at parent j, as a function of its child potentials. A Taylor contribution with m output arguments and child degree d, where 2\le m\le d\le4, assigns its d occurrence slots to nonempty blocks B_1,\ldots,B_m and uses the random functions

The directions in this derivative are supported at their indicated child labels; repeated labels are allowed. Multilinear substitution gives the complete polarized covariance coefficient of G_1,\ldots,G_m, with its Taylor factors and the single common weight \1_{\{\diam(j_1,\ldots,j_m)\le H\}}.

Discard size orders at least five, at cost O_H(z^5), and connect child slots when their enlarged stencils can share environment variables. A singleton centered child has zero expectation even conditional on the fresh priorities: the Taylor kernel depends on those independent priorities and takes the child function as a multilinear argument. A singleton mean costs size order two. Two components containing means only cannot create a lower-order term through random allocation. At the only relevant order their output is the linear map of Lemma 3.2, which is independent of priorities; its selection indicators are reference auxiliary variables, integrated by Q. In particular its charges are deterministic when its inputs are deterministic. We are therefore left with two distant components of bounded diameter and size order two each.

Every output argument receiving a child slot lies in that slot’s bounded stencil. These components consequently partition the relevant output arguments into two distant groups as well: each block B_a in (5.15) lies wholly in one component. Apply (5.9) to the two finite lists of output arguments G_a, with separate insertion indeterminates u_a. The groups are independent in the environment, including their fresh priorities. For their separate charges the formal identity

has zero coefficient involving variables from both groups, regardless of their separate means. Thus the unmixed contribution cancels for each fixed output tuple and each child-block partition, before any sum over output labels. Repeated child labels merely repeat derivative directions inside a group; no within-group independence is used. The remaining terms have a cross factor and hence an O_L(H^{-u}) bound from Proposition 2.5. The strip therefore costs O_L(H^{1-u})z^4=o_H(1)z^4. Keeping the covariance subtractions with each coefficient is essential here and is permitted because the cutoff depends only on the tuple of labels.

Finally apply Corollary 4.3. For each fixed output tuple, (5.3) is a sum of bounded multilinear expressions in at most four output potentials, including products of separately taken expectations. The charge bound on its fixed-H support makes the kernels uniformly bounded in s. The corollary applies to the finite union of all determining stencils, including their status checks. It therefore replaces the regular Taylor output by the actual output and removes child degrees above four with total error O_H(X^5)=O_H(z^5). In particular the highest intermediate degree is 4\cdot4=16<P=40, so the finite moment assumption suffices. This proves (5.13).□

Proof. Combine Lemmas 5.2, 5.3, and 5.4.□

Proof. There are two bonds per lattice site. Periodicity of the allocation therefore gives \sum_e\alpha_{0,e}=2s_0^2. Since Q(b_e)=1,

Independence of distinct bond signs and \sum_i\alpha_{i,e}=1 give the exact identity

Taylor expansion of the smooth function K proves the claimed asymptotics and derivative. The finite number of bounded bond functions in a cell gives the bounds on X_0 and Y_0. Lemma 5.2 gives the assertion about v_{H,0}.□

Proof. We first choose constants so that the moment bounds propagate whenever the mean remains between the walls m=\pm Dv. In the estimates of Proposition 4.1, write C_1 for a common constant and retain its contraction constant \rho<1. Choose A large enough for the initial X_0/\sqrt{v_0} bound and so that

Choose D next, with

Choose T next so that, for every subsequently fixed H, the initial ratios m_0/(Dv_{H,0}) at h=-T\theta^2 and h=T\theta^2 are strictly below -1 and above 1 for small \theta. Lemma 6.1 permits this choice independently of H: the leading denominator is 2Da^2\theta^2, whereas the leading numerator divided by \theta^2 is an affine function of h/\theta^2 with positive slope 2s_0a. Finally choose B sufficiently large that

and Y_0\le Bv_{H,0} for |h|\le T\theta^2 and sufficiently small \theta, for every subsequently fixed H. This last choice is possible from Y_0\le C_{s_0}(1+T)\theta^2 and v_{H,0}/\theta^2\to2a^2. The order so far is A,D,T,B.

For data satisfying (6.4), Lemma 5.2 gives

where the implicit constant includes A^3. Use Proposition 5.5 with z=\sqrt v. Its constants A,B are now fixed. Choose H sufficiently large that the e_Hv^2 error is smaller than one quarter of the negative leading term, using the positive limit 2\pi\log L of the shell sum. Then restrict v to a sufficiently small interval, also within the smallness requirement of Proposition 4.1, so that (6.9) and the O_H(v^{5/2}) remainder yield

Here c,C_2>0 are fixed and the smallness interval is uniform in the scale. Positivity follows, for example, by taking C_2v<1/2.

The X estimate now gives

The last inequality follows from (6.6) and v'/v\to1. Similarly,

by (6.8). At the upper wall, the mean estimate gives

at the lower wall it gives m'<-Dv'. We decreased the smallness interval if necessary to use (6.7) and (6.10). Thus the walls point outward, while both norm bounds propagate up to and including the first exit step.

We now restrict \theta so that the initial interval [-T\theta^2,T\theta^2] has all the properties fixed above and lies in the uniform smallness range just obtained. On this interval v_0=2a^2\theta^2+o(\theta^2) uniformly. The initial X bound is covered by A, because changing h by O(\theta^2) does not change the leading O(\theta) bound on X_0. Thus the complete parameter order is A,D,T,B,H, followed by the small interval for \theta.

All moment data are continuous in h at every finite number of steps. To check the only possible discontinuities, fix h and couple the constructions for nearby parameters with the same environment, priorities, and threshold variables. At a threshold comparison, the threshold is independent of the data being compared and has a continuous distribution. Equality therefore has probability zero. At finitely many steps only finitely many comparisons enter any specified local stencil, so the corresponding potentials are almost surely continuous at the fixed h. The crude real-log bounds of Proposition 3.3 dominate them on every compact initial parameter interval by a finite deterministic bound at each fixed number of steps. Dominated convergence applies to X,Y,m,W and to the finite polynomial v_H.

Continuity first supplies a closed initial subinterval I_0 on which -1\le m_0/(Dv_0)\le1, with its two endpoints mapped to -1 and 1. Suppose I_k has been chosen so that all parameters in it satisfy (6.4) through time k, and the endpoints lie on opposite time-k walls. The estimates just proved give positive v_{k+1} and the two norm bounds throughout I_k. Its endpoints map strictly outside the opposite time-(k+1) walls. The intermediate value theorem therefore gives a closed subinterval I_{k+1}\subset I_k on which -1\le m_{k+1}/(Dv_{k+1})\le1, with endpoints on the two walls. For example, take a last crossing of the lower wall before a first subsequent crossing of the upper wall. The preceding-time bounds remain valid on this smaller interval. Compactness gives a parameter h\in\bigcap_{k\ge0}I_k. No monotonicity of an iterated coordinate in h is needed.

For this parameter, (6.10) holds at every step. Taking reciprocals yields

and proves (6.5). The other conclusions follow from (6.4) and v_0=O(\theta^2).□

Proof. Activate the finitely many relevant marks in deterministic order. For fixed \eta and exterior data, conditioning on all previous marks charges the first instability at x to p_{x,k-1}(\eta) times a pivotal possibility event with that mark omitted. If adding its square first permits two test values, choose a witnessing assignment and revert the new square, retaining all the other assigned values. The resulting pair differs only in that square. Every passage required outside it by the geometric implication is consequently possible using only the previous marks. We may enlarge these to all other marks in their respective ring neighborhoods.

Consider a retained family of rings with checks T_1,\ldots,T_m. For fixed environment, condition off the first ring’s eroded chart W_{1,k} and on the marks whose squares do not meet its spin support. The other ring checks are measurable under this conditioning; the remaining spins have the specified conditional Gibbs law, and the remaining marks retain their independent law. Its factor is at most B^+_{k,T_1}(\eta). Remove each ring in turn. Pointwise in the original exterior data this proves

The enlarged ring neighborhoods are disjoint and avoid the environment stencil of p_{x,k-1}. All factors on the right are therefore functions of disjoint sets of the underlying independent environment variables. Taking their expectation, including that of the inserted mark, gives

This is why conditional exterior suprema, and the omission of rings near the inserted square, are needed. A product of unconditional quenched check probabilities would not justify this equality.

Retain rings of ratio a with successive inner radii in ratio Ga in each of the three ranges in (7.4), illustrated in Figure 2. A fixed number of omissions handles their endpoints, recenterings, and transitions. By (7.3), the product in (7.11), apart from C_LX_{k-1}, is at most

If a range has too few scales for a retained ring, its bounded factor loss is included in C_T.

It remains to sum over edit locations. Normalize a_T=1 and suppose b<1. The centers with dyadically specified truncated distances u,v occupy area at most C_Tuv: near a vertex, the portion of side at distance O(v) has length O_T(v) and its u-neighborhood has that area; away from vertices use the finite total side length. The class u=v=1 covers the remaining interior centers. Since the center grid has spacing comparable to b, enlarging each class by one grid cell gives at most C_Tuv/b^2 centers. Its contribution is bounded by

Put d=\min\{p_4-2,p_h-1,p_1\}>0. The three base ratios in the last expression have product b; each dyadic class therefore contributes at most C_TX_{k-1}b^d. There are O_T((1+\log(1/b))^2) classes. Absorb this factor into b^{d/2} and take c=d/2. For b\geq1, only O_T(1) grid centers have squares meeting the test support, and the crude pivotal bound one gives C_TX_{k-1}. This proves (7.9), after restoring a_T. Finally T^+-T\leq D_T, and partial activation only reduces the permitted mark parameters.□

Proof. We prove both assertions by strong induction on dyadic ranges of r/s_k. This induction includes all levels and all frames at once. At a terminal ratio, M_0s_k\leq r<M_0s_{k+1}, only a bounded number of level-k potentials meet the free annular chart. Put S equal to the sum of their norms. Formula (7.5) implies, for every exterior assignment,

Together with the pure bound q_i^0 this proves the terminal plain estimate. Since s_k/r\geq(M_0L)^{-1}, increasing the fixed constant puts it in the form (7.14).

For a nonterminal plain check, perform one step in its free chart. Conditional on the remaining exterior, the new interior has its level-(k+1) specification. Reversal and (7.7) bound its old probability by the new plain probability plus the probability of instability under marks from step k. The new plain check has ratio r/s_{k+1}=(r/s_k)/L and is already covered by induction. Every ring used to bound the instability has inner radius at most r/2 and is evaluated at level k+1, with the single-level possibility law there. Its ratio to the level spacing is strictly smaller than r/s_k, and it is monitored because the first retained radius was chosen above a sufficiently large multiple of s_{k+1}. Its eroded collars remain in the region where the new specification is valid. Thus the inductive possibility bound supplies the hypothesis of Lemma 7.1. That lemma costs C X_k(s_{k+1}/r)^c, which is a fixed multiple of X_k(s_k/r)^c. Adding it to the inductive plain estimate gives (7.14).

Once the plain estimate at a given ratio has been proved, bound its possibility increase using marks from step k-1. Its pivotal rings are now at level k, with inner radii at most r/2, so their possibility bounds belong to strictly smaller dyadic ranges of r/s_k. Lemma 7.1 gives the last term in (7.15). This also proves the terminal possibility estimate. If no monitored ring fits between the required first radius and the cutoff, the test size and edit radius differ by a bounded factor; the crude bounded count of candidate marks gives the same estimate without a recursive call. This starts the induction.

To close it, the geometric spacing gives

Choose C_{\rm boot} to cover the terminal comparison and the two one-level additions, and then choose \varepsilon_* so that this weighted sum and the additional single-level term are smaller than \min_i\tau_i. In each inductive call only these fixed tolerances were used to obtain the pivotal power. This proves both estimates without any assumption about a configuration edited at several levels.□

Proof. Couple each successive level by the reversal in Proposition 3.5. After averaging over the environment, the probability that a test changes at step j is at most

This follows from Lemma 7.1 and Proposition 7.2; if the test is no larger than the edit square, use the crude candidate count. Sum over steps and the fixed family. Equation (7.16) makes the sum vanish. At level K only a bounded number of potentials remain on the torus, so their expected total norm is O_M(X_K). The same density comparison as in the terminal bootstrap couples this level to the pure torus with expected total variation tending to zero. Gluing the successive couplings and using the union bound proves (7.17). Only the errors of individual reversals were summed; no probability estimate was imposed on the union of their marks.

The level-zero quenched probabilities depend only on the original bond environment \omega. Averaging the bounds over the auxiliary algorithm choices therefore leaves exactly the expectation in (7.17). Finally, choosing K by the first scale above a fixed multiple of \delta^{-1} keeps the physical period between two fixed bounds whose ratio is L.□

Proof. Choose the terminal spacing s_K comparable to r with a sufficiently small fixed proportionality constant depending on this geometry. Run the local construction through level K. The sum of the removed boundary thicknesses is O(s_K), so this choice leaves all prescribed buffers free. At each reversal use the conditional-supremum version of (7.10). For example, a measurable upper bound for the error is the sum over inserted centers of their mark parameters times the products of local ring suprema, with the factor one when only the crude count is used. These functions depend on local environment, and bound the error for every exterior assignment before any mixture of those assignments is taken. Summing them over steps and the finite family gives an envelope with expectation at most

At the terminal scale add C\min\{1,\sum_j\|f_j^{(K)}\|\} over potentials touching the remaining window. Their number is bounded, so this contribution has expectation O_{\mathcal T}(X_K). Conditional on its exterior, deleting those potentials gives the pure conditional law; integrating the exterior produces the stated mixture. Formula (7.16), with effective mesh r^{-1}, proves the vanishing expectation.

If every pure conditional law gives an event probability at least c_*>0, the perturbed probability is at least c_*-\mathcal E_r(\eta) for every initial exterior. Markov’s inequality then transfers any lower bound less than c_*. In particular this applies to the fixed compatible systems of buffered colored corridors and circuits in [22, Section 2, Proposition “Pure buffered RSW consequences”]. To state the result using only the original bond environment, average the envelope over the algorithm choices. The worst level-zero conditional probability itself depends only on the bonds in the window and its incident layer; hence its good-environment event is local. All estimates are uniform in the center, whereas their convergence rate may depend on the fixed geometry.□

Proof. For a specified center, pack separated rings of ratio a from a fixed multiple of r to a fixed fraction of R. At every fixed physical r>0, their inner lattice radii eventually exceed M_0s_0. Condition off their free charts as in (7.10) and use the plain bound of Proposition 7.2. Their environment neighborhoods are disjoint, so (7.11), without an inserted mark, yields

in the mesh upper limit, with p_4>2. The bound already includes the supremum over exterior data of the ambient domain. For fixed r, cover K_0 by an r-grid of O_{K_0}(r^{-2}) centers. Enlarge the inner squares and reduce the outer radius by fixed factors, so four passages about any center imply a grid check. A union bound gives C_{K_0,R}r^{p_4-2} after the mesh limit, which tends to zero. The finite test family is enlarged only after its mesh limit has been taken; lattice rounding is absorbed by the same fixed crops.□

Proof. We record the finite-volume form of Kramers–Wannier duality [18], including its homology convention; the four-sector matrix also appears in [4, Equations (4)–(5)]. For b\in(\Z/2\Z)^2, let

The high-temperature expansion gives

The low-temperature expansion of the dual spin system with holonomy \beta has domain walls of winding class (\beta_2,\beta_1). Each admissible wall configuration comes from two spin configurations, so

Substitution gives the sign (-1)^{\alpha_1\beta_2+\alpha_2\beta_1} in (8.5). Its prefactor follows from |E(\mathbb T_N)|=2N^2 and e^{-K_{e^*}^*}\cosh K_e=\sqrt{\sinh(2K_e)/2}. At K_e=K_0, the dual couplings equal K_0 and a(K)=1; hence \mathsf HZ(K_0)=Z(K_0). The displayed matrix satisfies \mathsf H^2=I.

For the ratio, expand each edge factor as

On a chosen open subgraph the spin constraints are consistent exactly when every open cycle has zero evaluation under \alpha. In that case there are 2^{k(A)} compatible spin assignments. Dividing the resulting sum by its untwisted version proves (8.6).□

Proof. Choose a fixed large integer M and torus periods N_k=Ms_k, so that every complete support and environment-dependence neighborhood of Proposition 3.3, through scale k, embeds in the torus. The primitive environment variables in each such neighborhood have their plane distribution, so each resulting potential has its plane law when expressed in local spin coordinates. Apply the scale construction with iid torus disorder, simultaneously in the four spin holonomies. Let \zeta denote the independent algorithmic choices used in this run. In every contractible neighborhood a flat twist has a spin trivialization; changes of trivialization on overlaps are constant spin flips. Every interaction is even, and the auxiliary variables are unchanged by spin flip. The local identities therefore hold in each holonomy, with the same scalar factors removed by centering. In particular, for a common positive factor c_k(\omega,\zeta) the resulting partition functions satisfy

where S_k=S_k(\omega,\zeta) can be the largest, over the four holonomies, of the sums of norms of the remaining potentials. There are only M^2 cells at the last scale. The trajectory bound X_k\to0 gives \E S_k\le C M^2 X_k\to0. Consequently

The partition vector itself does not depend on the algorithmic choices, so this conclusion also holds after retaining only the original bond environment.

The dual odds satisfy v_{e^*}^*=2/v_e. Thus duality sends (h,\xi_e) to (-h,-\xi_e), with the signs transported along the crossing-edge bijection. This transformation preserves their iid law. By Lemma 8.1, the normalized duality transform is the orthogonal matrix \mathsf H, and its pure target is fixed. Therefore (8.7) holds with -h_* as well.

Take \alpha=10, and let A_\alpha be the increasing event that some open cycle has odd evaluation under \alpha. Its pure probability q_N^0 satisfies

uniformly for large N. For the lower bound, a finite chain of pure FK rectangle crossings in a torus band constructs an open cycle winding once horizontally. For the upper bound, the corresponding construction of a closed-dual cycle winding once vertically prevents such an open cycle. The constructions use finitely many embedded rectangles of fixed aspect ratios and the uniform pure FK bounds recalled in [22, Section 2, “Pure buffered RSW consequences”]. They can be imposed successively with arbitrary induced boundary wiring; for a closed-dual crossing use the worst, fully wired primal boundary.

All ratios in (8.6) belong to [0,1], so the untwisted coordinate of \widehat Z is at least 1/2. Hence (8.7) implies, on the same original signs,

in probability, where q_{N,\omega}(h)= \phi_{\mathbb T_N,h,\omega}(A_\alpha). The two convergence statements hold simultaneously by a union bound. Querying every edge in (8.3) and using (8.4) gives q'_{N,\omega}(h)\ge q_{N,\omega}(h)(1-q_{N,\omega}(h)). If h_*\ne0, monotonicity and (8.8) keep this product bounded below on the interval between -h_* and h_*, with probability tending to one. Integration forces an endpoint difference bounded below by a positive constant, whereas both endpoints approach the same q_{N_k}^0. This contradiction proves h_*=0.□

Proof. Choose two separated buffered subbands of an annulus of fixed sufficiently large ratio a. The pure probability of simultaneous plus and minus spin circuits, one in each subband, is bounded below by a constant 2c_0>0, uniformly over pins outside the buffers; this is precisely [22, Section 2, Proposition “Pure buffered RSW consequences”]. Proposition 7.4 transfers this lower bound to the random spin law, with probability tending to one over the local environment and uniformly over the boundary pins. Apply it to Edwards–Sokal spins with the wired boundary cluster pinned plus. An ordinary open FK path has a constant spin. Such a path cannot cross both separating circuits of opposite colors. The common triangulation used for spin circuits causes no exception: a nearest-neighbor lattice path can cross a circuit only through a vertex of that circuit. The ordinary radial FK crossing probability is thus at most 1-c_0.

Defining goodness by this wired crossing probability makes it a function of the original local bonds alone. Its failure probability tends to zero after integrating out all algorithmic choices. FK monotonicity gives the assertion for h\le0 and for weaker wiring.

For completeness, this also rules out percolation at h=0. Pack concentric annuli with a fixed large geometric separation, so that their bond sets, including incident layers, are disjoint. Conditional on all edges off one annulus, its induced FK boundary partition is dominated by full wiring. The probability of an arm crossing m such annuli is therefore bounded, at fixed environment, by the product of their fully wired crossing bounds. Choose the first radius large enough that every bad-annulus probability is at most 1/2. Independence of the local bond sets bounds the environmental mean of this product by (1-c_0/2)^m. Letting m\to\infty in the wired infinite-volume limit proves (8.9).□

Proof. We work at x=0; the estimates are uniform over translates. For every vertex z\in B_{N+1}, pack m_N\ge c\log N separated annuli centered at z, with inner radii between N^{1/4} and a sufficiently small multiple of N^{1/2}. Their full annular graphs lie in B_{N^{1/2}}(z), and their bond neighborhoods are disjoint. Let \eta_N\to0 be an upper bound for their bad-environment probabilities, uniformly over all these radii and centers, provided by Lemma 8.3. Independence for the annuli around one fixed z gives

Let \mathcal G_N be the event that fewer than half are bad at every z\in B_{N+1}. A union bound, without any independence assertion between distinct centers, gives

Indeed m_N is bounded above and below by constant multiples of \log N, while \log(1/\eta_N)\to\infty.

On \mathcal G_N, multiplication of the good-annulus bounds gives, for some fixed \alpha>0,

For large N all these small boxes lie within B_{2N}. Induced boundary conditions are harmless because the annular bounds already use full wiring. The event \mathcal G_N works simultaneously for every h\le0 by monotonicity.

Fix such an environment and write

Choose an integer \ell uniformly between N/4 and N/2, and explore all ordinary open clusters in B_N that meet the layer \partial B_\ell. This determines the event defining q_\omega: every path from 0 to \partial B_N meets that layer. The exploration uses only ordinary edges in B_N, although its law is the marginal of the wired measure on B_{2N}. This is the layer exploration of [10, Lemma 3.2 and its proof].

An edge is queried only if an endpoint lies on the chosen layer or has an ordinary open path to it. For a fixed endpoint z, the probability, over \ell, that the layer is within distance N^{1/2} of z is at most C N^{-1/2}. Otherwise such a path forces an arm of length N^{1/2} at z. Thus (8.12) bounds the maximum averaged revealment by

Edges outside B_N have revealment zero. Their covariances are nonnegative, so retaining them in the derivative sum only strengthens the bound obtained from (8.3). Equations (8.3)–(8.4) yield, for all h\le0,

Moreover (8.12) at z=0 gives q_\omega(0)\le C N^{-\alpha}. For sufficiently large N, integrating the derivative of \log(q_\omega/(1-q_\omega)) over [-b,0] gives

Combining this estimate with (8.11) proves (8.10). In a larger graph, condition outside B_{2N}(x) and dominate the induced boundary partition by full wiring. This proves the final assertion, including infinite-volume limits.□

Proof. Lemma 8.3 and monotonicity give m_\theta(h)=0 for h\le0. Fix h=b>0. Planar FK duality replaces the edge odds by 2/v_e, so the dual environment has the law of the model at -b. In finite domains the dual boundary condition is dominated by full wiring. Taking limits, the dual of the infinite-volume wired primal law therefore satisfies the local annealed arm bounds of Lemma 8.4.

A closed-dual circuit surrounding B_M whose farthest distance from the origin lies in [R,2R) has diameter at least cR. It consequently contains an ordinary dual arm of length c'R from some dual vertex in B_{2R}, with absolute positive constants c,c'. There are O(R^2) possible vertices. Using (8.10) with any A>2 and summing over dyadic R\ge c''M gives

Here the arm bound is applied in a box of radius comparable with R, with a further collar of comparable width; its validity for arbitrary exterior boundary conditions was part of Lemma 8.4.

If every open cluster meeting B_M were finite, their union would be finite. It contains B_M, and its lattice edge boundary consists of closed edges. Connecting the union through the box, if necessary, does not change its exterior boundary. Its outer dual boundary therefore contains a closed-dual circuit surrounding B_M. The complement of the event in (8.15) thus forces some vertex of B_M to belong to an infinite open cluster. For large M this has positive joint probability. The joint environment–wired FK law is translation invariant, whence

This proves positivity for every b>0.

Finally, the left side of (8.14), viewed as a function of \beta\ge0, is continuous and strictly increasing from zero to infinity. Let \beta_* be its unique solution, and set

Then v_\pm=\sqrt2e^{\pm\theta_\epsilon}. Comparison with the constant couplings gives K_0/(1+\epsilon)\le\beta_*\le K_0/(1-\epsilon), so \theta_\epsilon\to0 as \epsilon\to0. The model at \beta_* therefore falls within the result just proved and has zero plus magnetization. The same holds below \beta_* by monotonicity. For \beta>\beta_* both log odds increase, and

The bonds at \beta dominate those of (8.1) with parameters (b,\theta_\epsilon) in the same sign environment. Their plus magnetization is consequently positive. This proves \beta_*=\beta_c(\epsilon) with the normalization and plus boundary convention of Theorem 1.2.□

Proof. On \{\varepsilon\le a/2\}, the second conditioning probability is at least a/2. Subtracting the two conditional probability fractions gives

Couple these two vectors maximally and glue this coupling to the two order couplings. Extend the outer vectors to samples with laws \lambda and \mu_\omega by their conditional laws given U and W. The outer vectors are ordered unless the two middle vectors disagree. On \{\varepsilon>a/2\} take any coupling with the same outer marginals. Markov’s inequality bounds this exceptional environmental probability by 2\E_{\mathrm{env}}\varepsilon/a, proving (9.1). All sets are finite. The coupling matrices and subsequent conditional probabilities can be selected measurably; for instance, fix an enumeration of feasible basic solutions of the finite coupling constraints and choose the first available one.□

Proof. Put the corridors, their reveal regions, and the finitely many strips inside a torus chart with a fixed positive external buffer. Choose a torus period allowed by Proposition 7.3. In this chart use the same bond environment as the target; complete the torus with independent bonds. Write E_h for the event that both cutting corridors have their prescribed lengthwise colored crossings, and write V for the vector of ordinary plus crossings of the strips on the torus. The pure buffered crossing bounds give

for all sufficiently fine meshes. Here a_h may depend on the fixed geometry. With q_{n,\omega} denoting the corresponding quenched torus vector law, Proposition 7.3 gives

At this stage the environmental expectation includes the independently added torus bonds.

We verify the two order comparisons needed for Lemma 9.1. In either torus, conditional on E_h, reveal the prescribed crossing nearest the exterior side of each corridor. The stopped reveal of [22, Section 6, Lemma “Reveals with a monochromatic facing boundary”; Section 7, proof of “Fixed-geometry boundary squeeze”] leaves only pins of the prescribed color facing the unrevealed center. The transcript already determines crossing existence, so no additional conditioning on E_h remains in that center. Disjoint reveal regions allow both explorations.

First compare the pure Dobrushin law in A_h^- with the conditioned pure torus. The revealed plus path is an upper plus pin against a lower free spin. It screens the plus arc of A_h^-; every accessible remaining boundary incidence is a lower minus pin against an upper free spin. Meetings of a cut and boundary give upper plus against lower minus. The minus reveal is outside A_h^-. Ferromagnetic boundary comparison therefore puts the pure-domain strip vector below p_n(V\in\cdot\mid E_h).

Next compare the conditioned random torus, as the lower law, with the target, as the upper law. They have identical couplings on every bond in their common free region. The minus path screens the target minus arc. The remaining boundary incidences, with upper entry first, are exactly

Each comparison follows by conditioning a free spin to its extremal value and using the nearest-neighbor ferromagnetic order. This also explains why no comparison between different random bond strengths is needed. The source’s geometric incidence check, including the corridor caps, shows that these are all incidences encountered by the common components containing the strips. A nearest-neighbor edge cannot cross the interior of a separating triangulated diagonal. Virtual plus values outside D_n give the same order for portions of a strip that exit and reenter the target. Thus q_{n,\omega}(V\in\cdot\mid E_h) is below the target strip vector.

Apply Lemma 9.1 with a=a_h and (9.3). It gives (9.2), extending the strip vectors to the complete spin samples with their conditional laws. Finally average over the extra torus bonds while retaining the original environment. The conditional pure marginal is unchanged, and the target marginal never used the extra bonds. Interchanging the signs proves the other assertion.□

Source and scope. The first assertions are [22, Section 7, “From fixed strip tests to limiting barriers”]. Uniform boundary parametrization gives uniform convergence of the normalized conformal maps on the closed disk, including the two marks. Mapping one disk \mathrm{SLE}_3 by these maps proves the displaced continuum convergence. For fixed h, only pure convergence in an ordinary polygonal approximation is used.

The finite family is obtained by covering favorable simple comparison chords with open neighborhoods described by rational polygonal strips, transverse gates, and strict crossing margins. Compact middle subarcs are in D; the two short terminal pieces tend to the marks. A countable such cover has a finite subfamily carrying arbitrarily large probability. Its entire list is fixed before taking the lattice limit; the sampled pure chord may select a strip only after the joint vector coupling has been made. Following the signed side of the lattice interface supplies the crossing with O(\delta_n) drawing error.

The last assertion is the winding and boundary-order argument of [22, Section 7, “Forbidden sides, order, and identification of the trace”]. A signed crossing in one of these strips blocks a target contour from the corresponding side. To include the open signed boundary arc, erase loops from the crossing and close it outside D_n. In the fixed \Psi coordinates, take radial connectors from its gates to a circle \rho=\rho_*>1 and the circular arc on the signed side, where \rho_* is independent of h,n. At fixed h, the added path has positive clearance from \overline D and hence lies outside D_n for large n. Include this threshold in the diagonal choice. The target marks are on the zero-winding side of the closed barrier. As the strip widths and terminal errors vanish, the gate centers tend to the marks, while the fixed-radius closure stays a positive distance beyond every compact subarc of the signed boundary. The winding exclusions therefore hold on every compact neighborhood of the forbidden side or boundary arc, giving (9.4). The argument uses only the deterministic geometry and noncrossing of opposite signed paths with the target contour; it applies to the couplings in Proposition 9.2.□

Proof. This is [22, Section 7, Lemma “Ordered chords with equal laws”]. Here is the argument. A point of P(L)\cap M(U) could be connected inside this union to both signed boundary arcs. Removing loops gives a crosscut between those arcs that avoids K. Its endpoints alternate with a,b, so it separates a from b, contradicting the connectedness of K. Hence P(L)\subset P(U)\cup U. No point of U can belong to the open set P(L), since every neighborhood of an interior point of U meets M(U). Thus P(L)\subset P(U).

Use a fixed Schoenflies homeomorphism to the disk and take the planar areas of the images of these two plus sides. They are bounded, ordered random variables with the same distribution, so their difference vanishes almost surely. If the inclusion of plus sides were proper, take z\in P(U)\setminus P(L). It either lies in M(L), or lies on L. In the latter case a neighborhood of z inside P(U) meets M(L), by the two-sidedness of a Jordan chord. In either case the difference contains a nonempty open subset of M(L), and hence has positive area. The plus sides are therefore equal, and so are their interior boundary chords. By (9.4), K lies on this chord. The inverse chord parametrization maps it to a connected subset of [0,1] containing both endpoints, which is all of [0,1].□

Source and interpretation. This is [22, Section 7, Lemma “Progress coordinates and the full curve metric”], with its passage convention of disjoint parameter intervals. Extend the inverse parametrization of G continuously to an ambient neighborhood and compose it with g_n to obtain u_n. When B_n\to0, the running maximum of u_n, normalized to endpoint values 0,1 and perturbed by a vanishing strictly increasing linear function, is an increasing homeomorphism uniformly close to u_n. Uniform continuity of G gives the asserted metric convergence. The other implication is the companion’s deterministic passage argument. In the random statement, compact exhaustion of D and deterministic lower bounds on R_\eta on events of increasing probability reduce its random centers and radii to (9.5). Six passages imply four, so the latter condition suffices. No bound on target path lengths is assumed.□

Completion of the proof of Theorem 1.2. Fix \epsilon in the interval already chosen in Sections 7 and 8. Begin with any subsequence of the given approximations, and pass further so that the designation of I_+ is fixed. We will construct, along this subsequence, deterministic comparison laws R_n and couplings such that

where R_n\Longrightarrow\mathrm{SLE}_3(D;a,b) in the curve metric.

Choose \eta_j\downarrow0 and the displacements and finite strip families of Lemma 9.3. For each fixed j, apply Proposition 9.2 to the two entire families. Choose increasing deterministic thresholds n_j so that for all n\ge n_j the geometric placements are valid, the pure comparison laws have their required approximation bounds, and each strip-transfer failure probability is at most \eta_j. The threshold may absorb the reciprocal of a_{h_j} from the conditioning estimate. Put

with any definition before n_1. This deterministic index tends to infinity, as slowly as necessary. Let L_n and U_n be the two pure comparison chords at displacement h_{j(n)}, and denote the law of L_n by R_n. Their deterministic marginal laws converge to the same \mathrm{SLE}_3(D;a,b) law.

For each n, glue the two strip-transfer couplings over the common pair (\omega,\Sigma_{n,\omega}). Equivalently, disintegrate each comparison sample conditional on that pair and then sample both conditional kernels. Each comparison marginal conditional on \omega is still its prescribed pure law. With probability tending to one, the target has both signed barriers selected from the finite lists by its two comparison chords.

All target traces lie eventually in one fixed compact planar set. Indeed uniform boundary convergence bounds \partial D_n, and the bounded components it encloses are bounded by the same large disk. The space of nonempty compact subsets of this disk is compact in Hausdorff distance. The ambient curve space is complete and separable: rational polygonal paths are dense, and from a Cauchy sequence one can select a subsequence with successive curve distances less than 2^{-j}. Recursively choose parametrizations of its representatives with successive uniform distances less than 2^{1-j}. They converge uniformly to a continuous path, which represents the limit of the original Cauchy sequence. Thus this space is Polish and the usual weak-convergence and representation theorems apply. The comparison chord laws are tight in this space. Along any further subsequence, pass again to a joint limit (K,L,U) of the target trace and comparison chords. The limit K is connected, contains a,b, and lies in \overline D; connectedness and endpoint inclusion pass under Hausdorff convergence, and points at positive distance outside \overline D are eventually outside D_n by the boundary winding argument. Lemma 9.3 gives (9.4). Since L and U have the same simple chord law, Lemma 9.4 gives

Represent this joint convergence with almost-sure convergence of the trace and comparison variables, and lift it to the target paths by their original conditional laws given those variables. At each mesh the target configuration space is finite, so this lift is immediate. It preserves every target passage probability and requires no prior tightness of the target paths in the curve metric. The marginal estimate Corollary 7.5 is precisely (9.5) for these target curves, including after this lift. Lemma 9.5 therefore gives d_{\mathrm{curv}}(\Gamma_n,L)\to0 in probability in the representation. The represented L_n already converges to L in that metric. The triangle inequality yields d_{\mathrm{curv}}(\Gamma_n,L_n)\to0 in probability. This statement depends only on the joint law of the pair at each n, so it holds in the original couplings as well. Since every further subsequence admits this argument, it holds along the whole chosen subsequence. Boundedness then proves the expectation assertion in (9.6).

Let \pi_{n,\omega} be the conditional joint law of (\Gamma_n,L_n) in that coupling, and write \overline d=\min\{1,d_{\mathrm{curv}}\}. Every function F in the defining class for d_{\mathrm{BL}} satisfies

The right side is independent of F, so taking its supremum and then environmental expectation gives

This step uses the conditional pure marginal in (9.6); its annealed counterpart alone would not give (9.7). The deterministic laws R_n converge in the same separable curve space to the required SLE law, and hence their bounded-Lipschitz distances to that law tend to zero. The triangle inequality and Markov’s inequality prove, for every \eta>0,

on the chosen subsequence. If convergence failed on the original sequence, a subsequence with the displayed probability bounded below would contradict the construction just given. This proves the full sequence assertion.

All couplings used only finitely many bond variables at each mesh; integrating out added torus bonds and algorithmic variables preserves the conditional laws given the original infinite environment. The bounded-Lipschitz distances are measurable, since Q_{n,\omega} takes only finitely many values. Finally restore the prescribed half-stubs and local drawing. Their matched displacement is O(\delta_n) and their orientation remains incoming to outgoing. Every displacement, strip, and comparison above was made in the original Euclidean embedding, and the interval of allowed \epsilon was fixed before choosing any target geometry.□