Proof. Edges lying along the spin boundary have both endpoints fixed at zero, so their interaction factors cancel from the expectation. Expand every other interaction as \[e^{b\cos(\theta_x-\theta_y)} =\sum_{j\in\mathbb Z}I_j(b)e^{ij(\theta_x-\theta_y)}.\] Integration imposes divergence conditions only at interior vertices. For the numerator these are the prescribed integer charges, with their sign determined by the current orientation; the denominator has zero divergence. There is no condition at a boundary vertex. Consequently one can construct a current carrying each charge along a path to the boundary, whether or not the total charge vanishes.

Subtract that fixed current in the numerator and rotate the remaining edge variables onto the dual edges. Every retained primal edge crosses exactly one edge of \(\mathcal G_n\). Every cycle of \(\mathcal G_n\) is generated by the elementary cycles around interior primal vertices, so the rotated zero-divergence current is a closed integral form. Integration along dual paths represents it as \(\nabla h/(2\pi)\), uniquely modulo one common height translation. The subtracted current gives the connection \(\alpha\). The factors \(e^b\) needed to replace \(I_j(b)\) by \(p_b(j)\) cancel. This proves the identity and also its positivity.

After holes are removed, any contractible cycle encloses no remaining charge, while a cycle winding around a hole has circulation equal to the total charge in that hole. Two integral flat connections with the same periods differ by an integral gradient: integrate their difference from a fixed vertex, using vanishing circulation for path independence. The last assertion follows by gauge change. ◻

Proof. We first give the Gaussian comparison that will implement an edge restoration. Let \(\Lambda\) be a lattice in a real Euclidean space, let \(z+\Lambda\) be a translate, and let \(T\) be positive definite. For \(Q\ge0\), \(t\ge0\), and a fixed real linear functional \(\ell\), put \[R(t)= \frac{\displaystyle\sum_{x\in z+\Lambda} e^{-\langle x,(T+tQ)x\rangle/2+\ell(x)}} {\displaystyle\sum_{x\in\Lambda} e^{-\langle x,(T+tQ)x\rangle/2}}.\] The relevant lattice-Gaussian inequalities originate in the work of Regev and Stephens-Davidowitz [16]. The covariance under the numerator law dominates that under the centered denominator law; see [14]. If \(m_{z,\ell}\) and \(\mathop{\mathrm{Cov}}_{z,\ell}\) denote the numerator mean and covariance, and \(\mathop{\mathrm{Cov}}_0\) the denominator covariance, differentiation gives \[\frac{d}{dt}\log R(t) =-\frac12\mathop{\mathrm{Tr}}Q\left( \mathop{\mathrm{Cov}}_{z,\ell}-\mathop{\mathrm{Cov}}_0+ m_{z,\ell}m_{z,\ell}^{\mathsf T}\right)\le0.\] For a proper affine subspace, first complete the square in its parallel span. The remaining offset is orthogonal for the current precision \(T+tQ\); the covariance on the parallel span is that of a translated, tilted lattice Gaussian with the restricted precision. Thus its covariance still dominates the centered covariance, while the raw mean \(m_{z,\ell}\) includes the fixed offset. The centered maximum of a Gaussian lattice sum gives the untilted bound \(R(t)\le1\).

To obtain the Bessel law, replace each primary edge by \(N\) independent integer increments with normalized weight \(q_N^{j^2}\), where \(q_N=b/(2N)\); their physical height increments are \(2\pi j\). Uniformly for \(u\) in compact real sets, \[\frac{\sum_{j\in\mathbb Z}q_N^{j^2}e^{uj}}{\sum_{j\in\mathbb Z}q_N^{j^2}} =1+\frac bN(\cosh u-1)+O_u(N^{-2}).\] Their sum tends to the probability weight \(p_b\), since its limiting moment generating function is \(\exp\{b(\cosh u-1)\}\). Matching endpoints, cycles, periods, and prescribed data gives parallel affine lattices for the numerator and denominator. Appending the residual coordinates \(Bh-v\) allows arbitrary real observation centers. Thus adding a quadratic penalty or a matching constraint is precisely the precision increase just considered. Centered covariance decreases under such increases by the same finite Gaussian comparisons; see [14].

All limits just described are on a fixed primary graph. A spanning forest expresses every height difference as a sum of full-chain increments, which have uniformly bounded exponential moments at each fixed real argument. The other normalized edge weights are at most one. This bounds the partition sums, their fixed real tilts, and any polynomial factors. A confined component constant is summed against its Gaussian penalty. Dominated convergence therefore passes the comparisons to \(p_b\), as in [15]. Equality limits use monotone convergence of the penalties; only nonempty affine fibers are used. For centered pin monotonicity, first use a finite pin penalty. The derivative of the logarithm of its expectation under an additional centered precision \(Q\) is \(\frac12\mathop{\mathrm{Tr}}Q(\mathop{\mathrm{Cov}}_{\rm unpinned}-\mathop{\mathrm{Cov}}_{\rm pinned})\ge0\). Then increase the pin penalty to infinity. The release of unconfined constants, including across cuts, is justified next. ◻

Proof. For a deleted edge \(e=(x,y)\), retain an independent chain with total increment \(J_e\in2\pi\mathbb Z\). Its normalized sum without matching is one. Restoring that edge imposes the equality \[J_e-h_y+h_x-\alpha_{xy}=0;\] the centered equality has \(\alpha_{xy}=0\). On the enlarged affine coordinates this amounts to increasing the coefficient of the square of the displayed residual to infinity. The prescribed-data fibers in numerator and denominator are parallel lattices. This remains true for relative data \(h_z-h_{z_0}=u_z\), since these are linear equality constraints. Lemma 8 gives the comparison. Integration of the matched chain contributes \(p_b((h_y-h_x+\alpha_{xy})/(2\pi))\), and all independent-chain normalizers cancel. For a real edge tilt, use \(J_e\) on the enlarged graph. On matching it is precisely the affine edge increment in the restored graph, so the tilted comparison applies with no additional scalar.

Here is the promised treatment of constants. Before comparing, add an auxiliary site penalty \(\delta\sum_xh_x^2/2\) and sum all translations. For a free component with \(m\) vertices, a translation orbit represented by \(h\), with mean \(\bar h\), contributes \[e^{-\frac\delta2\sum_x(h_x-\bar h)^2} \sum_{k\in\mathbb Z}e^{-\frac{m\delta}{2}(\bar h+2\pi k)^2}.\] Divide its translation sum by \[c_{m,\delta} =\frac1{2\pi}\sqrt{\frac{2\pi}{m\delta}}.\] Poisson summation shows that the quotient of the sum by \(c_{m,\delta}\) tends uniformly in \(\bar h\) to one and is uniformly bounded for small \(\delta\). The first factor is at most one and tends pointwise to one. Forest exponential-moment bounds again give domination, also for fixed real tilts invariant under the released constant. Thus the regularized sum divided by one \(c_{m,\delta}\) per free component tends to the quotient sum. Within each shifted-to-centered ratio, its component factors cancel. They cancel separately on the cut and restored sides, although those graphs can have different numbers of components.

On a component already fixed by absolute data or confined by observations, the auxiliary penalty is removed by ordinary dominated convergence, with no divergent factor. This also covers comparison of an unconfined ratio with a ratio in which new observations confine every component: cancellation is performed separately in each ratio. Semidefinite intermediate forms cause no difficulty, since the auxiliary penalty makes every sum proper before the limit. Finally take the chain limit at fixed primary graph as in Lemma 8. ◻

Proof. The cut comparison reads \[\frac{Z_{\rm restored}^s(u)}{Z_{\rm restored}^0(0_p)} \le \frac{Z_{\rm cut}^s(u)}{Z_{\rm cut}^0(0_p)}.\] Rearranging gives the claim. Every factor is positive and finite: on a finite quotient graph the forest bound proves finiteness, and all Bessel weights are positive. A joined component has only one global quotient, so the numerator sums the relative constants that the cut denominator quotients separately. ◻

Proof of the upper bound. Choose a smooth stream function \(\psi\) constant in a neighborhood of each boundary component, allowing different constants on different components, and put \(w=\nabla^\perp\psi\). Thus \(w\) vanishes near the boundary. Sample \(\psi\) on the faces of the height graph and take oriented differences across its edges. This gives an exactly divergence-free edge flow \(w_n\), supported away from the boundary, whose coefficients are \(n^{-1}w\) to first order. The form \(dv_0\) and \(\alpha_n\), integrated along lattice edges, have the same cycle periods. Their difference is therefore a real gradient. Consequently \[D_n'=\sum_e w_{n,e}(\nabla_e h+\alpha_{n,e}) =\sum_e w_{n,e}\alpha_{n,e} \longrightarrow \int_U w\cdot dv_0\] is deterministic.

Insert \(e^{D_n'}\) into the sector numerator. Partition \(U\) into dyadic squares of side \(d\), with \(d\) fixed and aligned with the boundary, and cut all edges between distinct squares. In applying Lemma 9, retain an independent Bessel increment on each cut edge and place that edge’s tilt on this increment. Restoring the edge identifies the increment with the affine height difference, so the statistic has not changed. The normalized tilted sector sum is bounded above by the resulting product of centered Laplace transforms. On each square the connection is an integral gradient and disappears by reindexing heights. Since the tilt is on the affine differences, this reindexing introduces no scalar tilt factor.

Here is the limiting bound on that product. On the internal edges of each square replace the coefficients of \(w_n\) by \(n^{-1}\) times one constant vector sampled there, and set the replacement to zero on cut edges. For fixed \(d\), the sum of squared errors on cut edges is \(O_d(n^{-1})\): there are \(O_d(n)\) such edges and every coefficient is \(O(n^{-1})\). The remaining squared error has a limsup tending to zero as \(d\downarrow0\), by smoothness of \(w\). The free-square Laplace limit in Theorem 4 applies to the constant-vector tests. Their total limiting log transform is \(a/2\) times the corresponding Riemann sum for \(\int_U|w|^2\).

For completeness, the errors can be kept inside the transforms. Apply Hölder’s inequality with exponent \(p>1\) to the constant-vector part. Bound the transform of the error, at the conjugate exponent, by freeing all Bessel increments. For every fixed tilt multiple, \[\log\mathbb E\exp\!\left\{\sum_e r_e J_e\right\} \le C\sum_e r_e^2\] when the \(J_e\) are independent increments of law \(p_{b_c}\) in height units and the \(r_e\) are uniformly bounded. This follows from \(\mathbb Ee^{tJ_e}=\exp\{b_c(\cosh(2\pi t)-1)\}\). First let \(n\to\infty\), then \(d\downarrow0\), and finally \(p\downarrow1\). We obtain \[ \limsup_{n\to\infty}\log F_{U,n}^s \le -\int_Uw\cdot dv_0+\frac a2\int_U|w|^2. \tag{10}\]

The fields \(w\) just used are dense in the orthogonal complement of gradients in \(L^2(U;\mathbb R^2)\). Indeed, extend a field in that complement by zero outside \(U\). Its distributional divergence vanishes, so in a larger simply connected box it has an \(H^1\) stream function. This function is constant on every complementary component of \(U\). Subtract a smooth function having the same constants near these components. The remainder has zero trace on \(\partial U\), and \(H^1_0(U)\) approximation now proves the density assertion. The gradient space is closed by the Poincaré inequality modulo constants. If \(P\) is the orthogonal projection onto its complement, then \(\mathcal M(U)=\|P\,dv_0\|_2^2\). Optimizing Equation (10) over the dense class, with optimizer \(a^{-1}P\,dv_0\), gives the required upper bound. ◻

Proof. At each fixed \(d\), Theorem 7 identifies the limit with the Gaussian determinant in Equation [eq:source-1]. We prove that this determinant tends to one. Equip the component \(H^1\) spaces with the norm \(\|u\|_d^2=\mathcal T_d(u)\), and write \[V_Q=\{u:u=0\text{ on }Q\}^{\perp_{\mathcal T_d}},\qquad C_d=\Pi_H|_{V_p},\] where \(\Pi_H\) is orthogonal projection onto \(V_H\). The letter \(Q\) in this definition can denote a pin region; the observation cells remain the squares in the sum defining \(\mathcal T_d\).

Rescale space by \(d^{-1}\). The observation cells are now unit squares. In dimension two the gradient energy is unchanged by this rescaling, and cell Poincaré gives, with constants independent of \(d\), \[ c\|u\|_{H^1}^2\le \mathcal T_d(u)\le C\|u\|_{H^1}^2 \quad\hbox{in these rescaled coordinates}. \tag{11}\] There are \(N_d=O(d^{-2})\) cells. If \(f\in V_p\), then \(\mathcal T_d(f,\phi)=0\) for every \(\phi\) vanishing on \(p\). In particular \(f\) is form-harmonic off \(p\).

We record the cutoff estimate, including the observation terms. Take a Lipschitz cutoff \(\chi\) vanishing near \(p\), with transition width and Lipschitz constant bounded in cell units, and test with \(\chi^2f\). The gradient part satisfies \[\int\nabla f\cdot\nabla(\chi^2f) \ge \frac12\int\chi^2|\nabla f|^2 -C\int_{\{\nabla\chi\ne0\}}f^2.\] On cells where \(\chi=1\), the observation contribution is \(f_Q^2\). On transition cells its absolute value is at most \(C\|f\|_{L^2(Q)}^2\), by Cauchy–Schwarz. Thus the energy on the plateau is bounded by the \(L^2\) mass in a fixed enlargement of the transition layer. The test is also admissible at a free side, where no boundary value is prescribed.

To iterate the estimate, let \(E_r\) be the sum of cell energies at cell-center distance at least \(r\) from \(p\). Choose \(\chi\) zero up to distance \(r+2\) and one beyond \(r+6\). By Equation (11), enlarging the layer by a fixed number of cells gives \[E_{r+10}\le C(E_r-E_{r+10}).\] Iteration gives exponential decay in cell distance. The fixed positive separation of \(H\) and \(p\) is of order \(d^{-1}\) in these coordinates. Multiplying \(f\) by a cutoff equal to one near \(H\) and supported away from \(p\) therefore produces an extension of its \(H\)-data of energy at most \(Ce^{-c/d}\|f\|_d^2\). Orthogonal projection gives the least-energy extension of these data, so, after adjusting the constants, \[ \|C_d\|\le \delta_d,\qquad \delta_d=Ce^{-c/d}. \tag{12}\]

An operator-norm bound alone does not control the determinant. We also need a summable singular-value bound. For an integer \(m\ge1\), restrict \(f\in V_p\) by requiring its Neumann cosine coefficients on every unit cell, for frequencies \(0\le k_1,k_2\le m\), to vanish. The codimension is at most \(C N_dm^2\), and the cell spectral inequality gives \[ \|f\|_{L^2(Q)}\le C m^{-1}\|\nabla f\|_{L^2(Q)}. \tag{13}\] Choose nested neighborhoods \(\mathcal N_1\subset\mathcal N_2\subset\mathcal N_3\) of \(H\), each a union of cells and separated from \(p\). Leave fixed-width buffers, in cell units, between successive neighborhoods and between \(H\) and the relative boundary of \(\mathcal N_1\). The neighborhoods may meet free walls. Define \[E_j(f)=\sum_{Q\subset\mathcal N_j} \left(a^{-1}\int_Q|\nabla f|^2+f_Q^2\right).\] Use cutoffs equal to one on \(\mathcal N_j\), supported in \(\mathcal N_{j+1}\), and with uniformly bounded gradients. In both harmonic tests the function is the original \(f\). The cutoff estimate and Equation (13) give \[E_1(f)\le C m^{-2}E_2(f) \le C m^{-4}E_3(f) \le C m^{-4}\|f\|_d^2 .\] Finally choose \(\eta=1\) near \(H\), supported in \(\mathcal N_1\), with bounded gradient. Cell coercivity bounds the cutoff extension by \(\|\eta f\|_d^2\le C E_1(f)\). Minimality of \(\Pi_Hf\) then gives \(\|\Pi_Hf\|_d\le C m^{-2}\|f\|_d\). The fixed positive physical separation leaves room for these buffers when \(d\) is small.

The min-max characterization of singular values consequently gives \[s_j(C_d)\le \frac{C N_d}{j}\quad (j\ge C N_d).\] Together with Equation (12), splitting the sum at an index of order \(N_d/\delta_d\) yields \[\|C_d\|_{\mathrm{HS}}^2 \le C N_d\delta_d\longrightarrow0 .\] For small \(d\), \(\|C_d\|<1/2\), and hence \[-\frac12\log\det(1-C_d^*C_d) \le \|C_d\|_{\mathrm{HS}}^2\longrightarrow0.\] Components contribute independently, so the proof also covers all the fixed cut geometries in the statement. ◻

Proof. Gauge away the connection on \(A\). The ratio becomes a centered confined Laplace transform in finitely many cell averages, multiplied by the scalar from expanding the shifted penalties. Theorem 6 gives its Gaussian limit, with the constant integrated. The sampled centers converge to the cell averages of \(v_A\). Convergence of transforms also holds at these converging arguments: Hölder bounds from transforms at fixed nearby arguments give local equicontinuity. Gaussian completion of the square identifies the limiting ratio as \[\exp\{-I_d(v_A)/2\},\qquad I_d(v_A)=\inf_{u\in H^1(A)} \left\{a^{-1}\int_A|\nabla u|^2+ \sum_{Q\subset A}(u_Q-(v_A)_Q)^2\right\}.\] The determinant is the same in the two Gaussian integrals and cancels. Trying \(u=v_A\) bounds \(I_d(v_A)\) by its gradient energy. For a sequence with bounded values of this functional, cell Poincaré gives \[\|u-v_A\|_{L^2(A)}^2 \le C d^2\left(\|\nabla u\|_2^2+\|\nabla v_A\|_2^2+ \sum_Q(u_Q-(v_A)_Q)^2\right)\longrightarrow0.\] Weak compactness in \(H^1\) and lower semicontinuity now show that \(I_d(v_A)\to a^{-1}\int_A|\nabla v_A|^2\). Independent components give the final assertion. ◻

Proof. Throughout this proof, \(Z_G^s\) and \(F_G^s=Z_G^s/Z_G^0\) include the cell penalties at the fixed parameters \(n,d,v\), on whichever cut topology \(G\) is indicated. In particular \(F_U^s=R_{n,d}^s(v)\). We suppress these parameters until taking the ordered limits.

The cuts and their separating guards. Connect each hole to a distinct unused part of the outer boundary by disjoint simple arcs. These can be chosen successively: opening a channel from one hole to the outside preserves connectivity and merges those two boundary components. Choose positive clearances from nonincident boundaries and previously chosen arcs. Approximate the arcs within those clearances by simple rectilinear paths with dyadic vertices, retaining short normal segments where they meet boundary sides and removing loops. Their endpoints lie in interiors of boundary segments. Narrow tubular strips around these paths, with square corners and dyadic widths, can then be chosen mutually disjoint, together with wider disjoint strip neighborhoods.

Delete the edges crossing the two sides of each narrow strip. This isolates the strips and leaves a domain with every hole slit. Every resulting piece admits a single-valued lift of \(v\). The strips, their wider neighborhoods, and all later bands have fixed positive widths. Choose the dyadic observation cells to align with all their boundaries. Coherent rounding makes these cells arrays entirely within their respective pieces.

Consider one of the finitely many strip deletions, with all earlier deletions retained, and denote its narrow strip by \(S\). Call the configurations before and after this deletion \({\rm open}\) and \({\rm cut}\). Inside a wider strip patch \(W\), place a guard \(p\) consisting of bands on both sides of the narrow strip. The bands separate every edge to be restored from the artificial boundary of \(W\), and have positive buffers to both. At the two ends they reach the original free walls. Parallel collars and square corners provide the same construction at turns. Thus \(W\) has a common single-valued lift, before and after the deletion. Figure 1 illustrates the arrangement.

For prescribed absolute primary data \(u\) on \(p\), let \[K^s(u)=\frac{Z_{\rm open}^s(u)}{Z_{\rm cut}^s(u)}, \qquad d_0=K^0(0).\] Conditioning on the full guard separates the restored edges from everything beyond \(W\), since no observation cell crosses a guard division. Therefore \(K^s(u)\) is unchanged if we cut the edges across \(\partial W\) and compute it using \(W\) alone. The prescribed-data comparison of Lemma 9 gives \[\frac{Z_{\rm open}^s(u)}{Z_{\rm open}^0(0)} \le \frac{Z_{\rm cut}^s(u)}{Z_{\rm cut}^0(0)}, \qquad\hbox{hence}\qquad 0<K^s(u)\le d_0.\]

The loss inside the patch. Choose a union of positive-area boxes \(H\) containing all endpoints of restored edges and positively separated from \(p\), in the cut patch. Under its centered confined law, \[1\ge \mathbb E_{{\rm cut}\,W,0}\frac{K^0(u)}{d_0} \ge e^{-D_{n,d}(H,p)}.\] To verify the lower bound, compare the restored partition sums with guard data \(u\) and \(0\), then add the pin \(h_H=0\). Precision comparison decreases their ratio. With \(H\) pinned, restoration is a fixed scalar because all restored-edge endpoints are fixed. Dividing by the corresponding cut-patch data ratio gives the pointwise lower comparator \[\frac{K^0(u)}{d_0} \ge \frac{\mathbb P_{{\rm cut}\,W,0}(h_H=0\mid h_p=u)} {\mathbb P_{{\rm cut}\,W,0}(h_H=0\mid h_p=0)}.\] Averaging and using Bayes’ rule gives the displayed estimate. Lemma 12 shows that its lower bound tends to one, in the order \(n\to\infty\), then \(d\downarrow0\).

Both versions of \(W\) have lifts, and the cuts remove only zero-area interfaces. Lemma 13 therefore gives the same limit for their shifted-to-centered ratios. It follows that \[ \sum_u\frac{Z_{{\rm cut}\,W}^s(u)}{Z_{{\rm cut}\,W}^0} \left(1-\frac{K^s(u)}{d_0}\right)\longrightarrow0 \quad(n\to\infty,\ \text{then }d\downarrow0). \tag{6} \] The arrow means that the iterated limsup is zero. For clarity, the sum in Equation [eq:source-6] is exactly \[F_{{\rm cut}\,W}^s- F_{{\rm open}\,W}^s\, \mathbb E_{{\rm cut}\,W,0}\frac{K^0}{d_0}.\] This identity proves the assertion without any approximation of conditional laws at individual microscopic values \(u\).

Passing from the patch to the whole domain. Let \(G\) be the current topology with the narrow strip cut, and let \(G'\) be obtained by further cutting the edges across the artificial boundary of \(W\). At prescribed guard data, precision comparison gives \[\frac{Z_G^s(u)}{Z_G^0(0_p)} \le \frac{Z_{G'}^s(u)}{Z_{G'}^0(0_p)}.\] Here \(0_p\) denotes prescribed zero guard data; a denominator without a data argument sums over those data. Thus \[ \frac{Z_G^s(u)}{Z_G^0} \le \frac{\mathbb P_{G,0}(h_p=0)}{\mathbb P_{G',0}(h_p=0)} \frac{Z_{G'}^s(u)}{Z_{G'}^0}. \tag{15}\] This change of normalization has a controlled cost. Let \(R(u)\) be the centered restoration factor from \(G'\) to \(G\) at guard data \(u\). Directly summing partition functions gives \[\frac{\mathbb P_{G,0}(h_p=0)}{\mathbb P_{G',0}(h_p=0)} =\frac{R(0)}{\mathbb E_{G',0}R}.\] Choose a union of boxes \(H'\) containing the endpoints of all edges across the artificial boundary of \(W\), separated from \(p\). The preceding centered restoration comparison, now with \(H'\), implies \[\frac{\mathbb E_{G',0}R}{R(0)} \ge e^{-D_{n,d}^{G'}(H',p)}.\] Lemma 12 makes this interaction tend to zero. All observation cells remain within their components after the cut, and control their constants, so its hypotheses hold for \(G'\).

Multiply Equation (15) by the nonnegative quantity \(1-K^s(u)/d_0\) and sum. On \(G'\), the part outside \(W\) factors off as a shifted-to-centered confined ratio, at most one by Lemma 8. The resulting global loss is therefore at most \(e^{D_{n,d}^{G'}(H',p)}\) times the patch loss in Equation [eq:source-6]; it tends to zero. The same conclusion applies to the centered loss.

To finish, define these two losses on the current cut graph by \[L_s=\sum_u\frac{Z_{\rm cut}^s(u)}{Z_{\rm cut}^0} \left(1-\frac{K^s(u)}{d_0}\right), \qquad L_0=\mathbb E_{{\rm cut},0}\left(1-\frac{K^0}{d_0}\right).\] Summing the restored numerator and denominator gives the exact identity \[ F_{\rm open}^s=\frac{F_{\rm cut}^s-L_s}{1-L_0}. \tag{16}\] Since \(L_s,L_0\) tend to zero in the prescribed order and \(F_{\rm cut}^s\le1\), a strip deletion changes the confined ratio by a quantity tending to zero. There are only finitely many strips. After all deletions, Lemma 13 identifies the product ratio with the right side of Equation [eq:source-5]. The energies add over the pieces to \(\int_U|dv|^2\). Restoring the strips proves the proposition. The raw factors \(d_0\) cancel in Equation (16); no bound on their size is needed. ◻

Completion of Theorem 11. For every smooth \(g\) near \(\overline U\), combine Equation (14) with Proposition 14 to obtain \[\liminf_{n\to\infty}F_{U,n}^s \ge \exp\left\{-\frac1{2a}\int_U|dv_0+dg|^2\right\}.\] Restrictions of smooth functions from a neighborhood are dense in \(H^1(U)\), since \(U\) is Lipschitz. Let their energies approach \(\mathcal M(U)\). This proves the lower bound in Equation [eq:source-3], and the earlier upper bound proves the limit. Every use of a height or pin theorem was at fixed geometry and fixed \(d\); all subsequent limits were taken afterwards. ◻

Proof. Lemma 9 gives the prescribed-data comparison \[\frac{Z_{\mathrm{restored}}^s(u)} {Z_{\mathrm{restored}}^0(0)} \le \frac{Z_{\mathrm{cut}}^s(u)}{Z_{\mathrm{cut}}^0(0)}.\] Rearranging proves Equation [eq:source-8]. The comparison applies to relative guard data: they are affine constraints on height differences. If auxiliary confinement is used to perform the comparison, its divergent translation scalar cancels within each displayed ratio before confinement is removed. The two graphs need not have the same number of free components.

For the lower bound, work in the centered sector and impose \([H]\) in every indicated annulus. Adding these centered constraints to the restored data-versus-zero ratio decreases that ratio. Once they hold, choose the representative in each annulus to be zero on its flat end. The restoration factor is then independent of the guard data: its remaining variables are the relative integer translations and, in the exterior case, the unprescribed exterior heights. Let \(\mathbb P_{\ell,0}\) denote the free centered law in annulus \(\ell\). The comparison, after division by the cut data-versus-zero ratio, gives \[\frac{T_i^0(u)}{d_i} \ge \prod_{\text{incident annuli }\ell} \frac{\mathbb P_{\ell,0}([H]\mid u_\ell)} {\mathbb P_{\ell,0}([H]\mid[p])}.\] All seam endpoints lie in the chosen end regions; hence no other annular heights enter the restoration factor with these events imposed. Averaging over the independent cut-block laws yields Equation [eq:source-9]. ◻

Proof. Begin with one representative of the heights in each cut block. On the tree whose vertices are these blocks and whose edges are their interfaces, integer block constants modulo one common constant are in bijection with the integer differences along the tree edges. The bijection has no multiplicity factor: fix the root constant, then recover each other constant along its unique path from the root. The relative translations can therefore be summed independently at the interfaces.

Condition next on every guard datum. Within each annulus, its inner and outer configurations are conditionally independent. The two incident interface factors may consequently be integrated separately, giving the kernels in Equation (21). For several nests, put the exterior at the root of the block tree. Its different boundary portions need not be independent, so their integrations remain together in the single exterior kernel. This is exactly the formula asserted. ◻

Proof. The integral of \(T_i^s\) under the independent cut-block laws is \(Z_{\mathrm{restored}}^s/Z_{\mathrm{cut}}^s\). Dividing this identity by its centered counterpart proves Equation [eq:source-10]. The same identity is valid for a joint exterior kernel, without a claim yet about its average loss.

For a circular annulus of radii \(1\) and \(\lambda\), the minimum energy with unit period \(2\pi\) is \(2\pi\log\lambda\). The angular function attains it, and integration on concentric circles proves the matching lower bound. Restriction of admissible forms, together with \(B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon)\), gives \[\mathcal M(\lambda P\setminus P^\circ) =2\pi\log\lambda+O(\varepsilon),\] where the constant is independent of large \(\lambda\). For example, the circular annulus with radii \(1+\varepsilon\) and \(\lambda(1-\varepsilon)\) is contained in the polygonal one, which in turn is contained in the circular annulus with radii \(1-\varepsilon\) and \(\lambda(1+\varepsilon)\). The same bounds apply with \(\lambda^2\) to the union of two blocks. It follows that \[0\le \mathcal M(\lambda^2P\setminus P^\circ) -2\mathcal M(\lambda P\setminus P^\circ) \le C\varepsilon.\] The lower bound follows by restriction to the two cut annuli; their energies agree by scale invariance in dimension two. Theorem 11, applied at this fixed geometry, therefore makes the last factor in Equation [eq:source-10] converge to a number in \([e^{-C\varepsilon/(2a)},1]\). Changing the unit period’s sign does not change its energy.

Choose \(\varepsilon\) to make this error small, and then choose \(\lambda\) so that Proposition 15 makes the two centered pin interactions small. Equation [eq:source-9] and Equation [eq:source-10] now give the desired small loss in the mesh limit. With \(P\) and \(\lambda\) fixed, every regular interface is a refinement of one fixed geometry. The convergence theorems hold along every admissible mesh sequence, so there is one threshold \(R_0\) beyond which the inequalities hold at every aligned scale. There are only three sectors to consider. This proves Equation [eq:source-11]. ◻

Proof. Define \[L_i f(v)=\int t_i(u,v)f(u)\,dm_i(u).\] First propagate the constant density: \[q_1=1,\qquad \beta_i=\int L_iq_i\,dm_{i+1},\qquad q_{i+1}=\beta_i^{-1}L_iq_i.\] Inductively \(q_i\le2\), because \[\beta_i =1-\int(1-t_i)q_i\,d(m_i\otimes m_{i+1}) \ge1-2\xi,\qquad q_{i+1}\le M:=(1-2\xi)^{-1}<2.\] The same argument, with the kernels reversed, bounds by \(2\) any normalized density obtained by propagating \(1\) backwards through a finite number of them.

Propagate \(g\) using these same scalars, rather than renormalizing it at each step: \[f_1=g,\qquad f_{i+1}=\beta_i^{-1}L_if_i,\qquad c_i=\int f_i\,dm_i,\qquad E_i=f_i-c_iq_i.\] The scalar \(c_i\) is the ratio of the two integrals obtained by starting with \(g\) and with \(1\). Equivalently, it is the expectation of \(g\) under a normalized backwards density on \(X_1\), which is at most \(2\). Choose \(\delta>0\) with \(m_1(g<\delta)<1/4\). Every such density assigns at least half its mass to \(\{g\ge\delta\}\), so \[ c_i\ge\delta/2 \quad\text{for every }i. \tag{22}\]

For a mean-zero function \(E\), replace \(t_i\) in \(L_iE\) by \(t_i-1\). Since \(0\le(1-t_i)^2\le1-t_i\), the resulting operator has \(L^2(m_i)\)-to-\(L^2(m_{i+1})\) norm at most \(\sqrt\xi\). The exact recurrence for the centered remainder is \[E_{i+1} =\frac{L_iE_i-\bigl(\int L_iE_i\,dm_{i+1}\bigr)q_{i+1}} {\beta_i}.\] Using \(\|q_{i+1}\|_2\le2\), we obtain \[ \|E_{i+1}\|_2 \le\frac{3\sqrt\xi}{1-2\xi}\|E_i\|_2. \tag{23}\] The factor is strictly smaller than one. Also \(t_i\le1\) gives \(\|L_iE_i\|_\infty\le\|E_i\|_2\), and hence \[\|E_{i+1}\|_\infty \le\frac{3}{1-2\xi}\|E_i\|_2.\] Thus the remainders tend to zero in the last displayed norm. The last-coordinate density in the lemma is \[f_N/c_N=q_N+E_N/c_N.\] Equation (22) and the strict inequality \(M<2\) show that it is eventually bounded by \(2\). ◻

Proof. Put \(t_i=T_i^s/d_i\) for regular interfaces and \(g=T_0^s/d_0\) for the core interface. Lemma 16 gives \(0<g\le1\) and \(0<t_i\le1\). Proposition 18 gives the average-loss hypothesis of Lemma 19. Exact multiplication in Lemma 17 identifies its last marginal with the actual last-guard law. The fixed initial weight \(g\) may be arbitrarily small on some data; positivity is sufficient.

A lattice translation preserves all block laws and kernels. Changing the source sign negates the connection and heights. More generally, a connection with the same source restricts on the nest to the chosen connection up to an integral gauge change, which merely reindexes the guard data. Taking the maximum of the finitely many thresholds for the centered and two unit sectors proves the assertion. ◻

Proof. Set \(U(y)=\sum_i\sigma_iG_D(y,z_i)\), and rotate \(-2\pi\nabla U\) by a right angle, with the orientation chosen to give the indicated periods. Denote the resulting vector field by \(v\). It is a smooth affine gradient on the punctured square, is divergence free there, and is tangent to the outer boundary because \(U\) has zero Dirichlet trace. Thus \(v\) is an admissible trial field.

Near \(z_i\), write \[U(y)=\frac{\sigma_i}{2\pi}\log\frac1{|y-z_i|}+u_i(y), \qquad u_i(z_i)=\sigma_iR_D(z_i,z_i) +\sum_{j\ne i}\sigma_jG_D(z_i,z_j).\] The functions \(u_i\) are smooth and harmonic on fixed disjoint neighborhoods of the sources. On the boundary of a hole, the radial singularity contributes no normal component to \(v\). Its remaining normal component is bounded uniformly in \(\rho\) and has integral zero, since it is a tangential derivative of \(u_i\).

Let \(g\in H^1\) on the punctured square. Integration by parts, followed by subtraction of the mean of \(g\) on each inner circle, gives \[\left|\int v\cdot\nabla g\right| \leq C\rho\,\|\nabla g\|_2.\] Indeed, the \(L^2\) norm of \(v\cdot\nu\) on such a circle is \(O(\rho^{1/2})\), whereas the trace and Poincaré inequalities on the adjacent annulus of radii \(\rho\) and \(2\rho\) give \[\|g-\text{its circle mean}\|_{L^2(\partial B(z_i,\rho))} \leq C\rho^{1/2} \|\nabla g\|_{L^2(B(z_i,2\rho)\setminus B(z_i,\rho))}.\] The collars are disjoint. Minimizing \(\|v+\nabla g\|_2^2\) therefore changes \(\|v\|_2^2\) by at most \(O(\rho^2)\).

It remains to calculate \(\|v\|_2^2=(2\pi)^2\int|\nabla U|^2\). The outer-boundary contribution to Green’s identity vanishes. On the circle about \(z_i\), with \(\nu\) pointing into the hole, the integral of \(U\partial_\nu U\) is \[\frac1{2\pi}\log(1/\rho)+\sigma_i u_i(z_i)+O(\rho).\] The integral of the normal derivative of the regular harmonic part is zero; in particular it produces no logarithmically enhanced error. Summing over the circles gives Equation [eq:source-12]. These integrations may also be performed with weak Dirichlet data on the outer square. Alternatively, reflection across its sides and corners, away from the sources, justifies the boundary calculation directly. ◻

Proof. Bracket the moving holes by fixed, positively separated rectilinear holes with dyadic coordinates. To obtain strict eventual inclusions, start with circles of radii \((1-3\varepsilon)\rho\) and \((1+3\varepsilon)\rho\) about the limiting points, and approximate them and their centers by still smaller dyadic perturbations. The perturbations can be chosen after fixing \(\rho\) and \(\varepsilon\). For all sufficiently large \(n\), the rounded moving holes lie between these fixed approximations and contain their respective sources. The outer boundary remains the same square.

Use one connection before making any of these cuts. Its restrictions to the three exteriors have the same winding data. The cut comparison in Equation [eq:source-2] therefore brackets their sector ratios. Theorem 11 applies to each fixed polygonal exterior. Its continuum minimum energy is in turn bracketed by the energies for the corresponding circular holes: restriction of an admissible affine gradient preserves all the remaining periods. Changing each radius by a factor \(1+O(\varepsilon)\) changes the logarithmic term of Equation [eq:source-12] by \(O(\varepsilon)\). That equation now proves the claim.

Every domain used in this argument has positive passages and separated hole boundaries. Coherent rounding consequently preserves connectivity and the cycles defining the periods. In particular the argument uses only fixed-domain limits; it does not require uniform convergence under arbitrary motion of the lattice marks. ◻

Proof. For the centered calculation, Equation [eq:source-9] bounds the attachment loss by the sum of the flat-pin interactions in the incident last annuli. These interactions can be made as small as desired by the choices in Section 4. The bound depends only on these annuli, not on the exterior.

For the shifted calculation use the exact identity in Equation [eq:source-10], now for the last annuli together with their common exterior. Restoring them produces an exterior whose smaller holes have radius parameter tending to \(t/\lambda\). Before restoration, the factors are the \(k\) last-annulus sector ratios and the exterior ratio with holes of radius parameter \(t\). By Theorem 11, each annular energy is \(2\pi\log\lambda+O(\varepsilon)\). By Equation [eq:source-13], the restored and cut exterior energies have the same term \(4\pi^2H_D(\mathbf z,\boldsymbol\sigma)\). The difference between their radius terms is exactly \(k\,2\pi\log\lambda\). Consequently the logarithm of the restored sector ratio divided by the product of the cut ratios has absolute limiting error at most \(C(\varepsilon+\eta)\). The circular energy brackets make this error uniform over \(t\in[\eta/\lambda,\eta]\) and over sufficiently large fixed \(\lambda\).

Choose \(\varepsilon\) and \(\eta\) small, then choose \(\lambda\) large enough for the flat-pin interactions, and finally choose \(R_0\) large enough for the fixed annular geometry. These choices retain the regular-interface estimate Equation [eq:source-11]. The preceding centered estimate, the restored-to-cut identity, and the upper bound \(T_{\rm ext}^s/d_{\rm ext}\leq1\) now give Equation [eq:source-14], after making the preliminary errors small enough in terms of \(\xi\). Apply the same choices to the one-source configuration at \(0\).

The argument was given after extraction of a limit \(t\). Every subsequence has such a further subsequence, and the error bounds are uniform over its possible values. It therefore proves the stated limiting bounds on the full sequence. All height and pin limits used here precede variation of the fixed geometric parameters; no rate uniform in a shrinking physical radius has been assumed. ◻

Proof. Let \(F_{\rm nest}(R_N)\) denote the bare sector ratio of one whole free nest with a unit source, including its core. It is the same for all the translated sources, for either sign, and for the center reference. By Lemma 17, the partition function of the whole graph is the product of its cut block partition functions times its gluing-kernel expectation. After the kernels internal to each nest have been integrated, its last-guard law has density at most \(2\) relative to \(m_N^s\), for all sufficiently large \(N\), by Corollary 20.

Write \(e_s\) for the expectation of \(T_{\rm ext}^s/d_{\rm ext}\) under the product of these propagated last-guard laws, and \(e_0\) for the corresponding centered expectation. Equation [eq:source-14] and the product density bound give \[1-2^k\xi-o(1)\leq e_s\leq1\] in both sectors. The centered factor \(d_{\rm ext}\) cancels between the numerator and denominator, leaving the exact identity \[\mathbb E_n^0\exp\left(i\sum_i\sigma_i\theta_{x_{i,n}}\right) =F_{\rm nest}(R_N)^k F_{\rm ext}^s\,\frac{e_s}{e_0}.\] Taking logarithms, with \(\xi\) small at this fixed \(k\), gives \[\log\mathbb E_n^0\exp\left(i\sum_i\sigma_i\theta_{x_{i,n}}\right) =k\log F_{\rm nest}(R_N)+\log F_{\rm ext}^s +O_k(\xi)+o(1).\] The same calculation with the single center source expresses \(\log a_n\) using this identical nest ratio. Subtracting \(k\) times that expression cancels the microscopic term exactly. In Equation [eq:source-13], the common-radius logarithms also cancel. The remaining coefficient is \[\frac{4\pi^2}{2a}=\frac1{2K_*}, \qquad a=8\pi,\quad K_*=\frac2\pi.\] This proves Equation [eq:source-15] up to an error that can be made arbitrarily small.

For completeness, the choices of nests may depend on that accuracy. This is harmless: the quantity on the left is always the same exact observable. If its asserted convergence failed, a subsequence with a fixed positive error would, at a sufficiently high chosen accuracy, have a further subsequence with \(R_N/n\to t\) and a smaller error by the preceding calculation, a contradiction. ◻

Proof. The killed Green estimate in [14] gives local uniform convergence of \(G_n\) to \(G_D\) off the diagonal, with exactly the unscaled Laplacian convention used here. We explain how to recover the difference of the diagonals without specifying their common lattice constant.

Choose a small fixed box \(Q\) of displacement vectors about \(0\) such that \(x_n+Q\) and \(Q\) remain in \(D\) for all large \(n\). For lattice displacement vectors \(v\), set \[h_n(v)=G_n(x_n,x_n+v)-G_n(0,v).\] Both kernels have the same unit source at \(v=0\). Their difference is therefore discrete harmonic throughout \(Q\), including at \(v=0\). On a coherently rounded boundary of \(Q\), the off-diagonal convergence identifies its limit as \[G_D(z,z+v)-G_D(0,v) =R_D(z,z+v)-R_D(0,v).\] The discrete maximum principle bounds \(h_n(0)\) between the minimum and maximum of these boundary values, up to an error tending to zero. Shrinking \(Q\) and using continuity gives Equation [eq:source-16]. If uniform boundedness on an interior compact failed, a violating sequence would have a convergent subsequence of its locations. The convergence just proved would contradict that violation. The finitely many remaining lattice sizes cause no difficulty. ◻

Proof. Exponentiate Equation [eq:source-15], multiply by the deterministic normalizing factors, and use Equation [eq:source-16]. The terms \(\sum_iR_D(z_i,z_i)-kR_D(0,0)\) cancel exactly. For uniformity, a sequence of violations in a compact separated configuration space would have a convergent subsequence, to which the just-proved sequential convergence applies. ◻

Proof. For the single center source, the normalized exterior attachment has a positive lower bound in the shifted sector and is at most one in the centered sector. Its exterior sector ratio is also bounded away from zero by Equation [eq:source-13]: every subsequential hole radius lies in the fixed interval \([\eta/\lambda,\eta]\). The exact gluing identity therefore gives Equation [eq:source-18]. The compact subsequence argument makes the lower bound uniform for all sufficiently large \(n\).

Next add one regular annulus to a nest. Its sector ratio multiplies the nest ratio, together with the ratio of the shifted and centered attachment expectations. Normalize these two expectations by their common centered zero-data value. The centered expectation is at most one. Once the nest is sufficiently long, its last-data density is at most two, so Equation [eq:source-11] makes the shifted expectation at least \(1-2\xi\).

By Theorem 11 and the circular annular energy bracket, the annular sector factor at large lattice scales is bounded below by \[\lambda^{-1/8}\exp\{-O(\varepsilon)-o(1)\}, \qquad \frac{\pi}{a}=\frac18.\] Include in the fixed parameter choices that \(\lambda\) and then the lattice threshold are sufficiently large. Each sufficiently late extension factor is then at least \(\lambda^{-1/2}\). There are only finitely many earlier extension factors, all strictly positive. Their losses relative to this lower bound can be absorbed into a single \(c'>0\), valid for every interval of extension indices. Multiplication proves Equation [eq:source-19]. ◻

Proof. Write \(r=n|x-y|\). Choose \(c_E>0\) so small that nests of radius \(R\leq c_Er\), centered at \(x\) and \(y\) in lattice units, are disjoint and contained in the spin square. Decrease \(c_E\), if necessary, so that \(c_Er\leq\eta n\) for all locations under consideration.

If there is a largest \(j\geq0\) with \(R_j\leq c_Er\), cut out these two nests. The restriction of the magnetic sector to each nest has its single unit source. The cut comparison and the upper bound one for the remaining sector ratio give \[\mathbb E_n^0e^{i\sigma\theta_x+i\tau\theta_y} \leq F_{\rm nest}(R_j)^2.\] Equations [eq:source-18] and [eq:source-19] yield \[\frac{\mathbb E_n^0e^{i\sigma\theta_x+i\tau\theta_y}}{a_n^2} \leq C\lambda^{N-j}.\] The maximality of \(j\) gives \(R_j>c_Er/\lambda\), while \(R_N\leq\eta n\). Since \(R_N/R_j=\lambda^{N-j}\), this is at most \(C_E n/(1+r)\), after increasing the constant.

If no such \(j\) exists, then \(r<R_0/c_E\). Use the bare expectation bound by one and Equation [eq:source-19] with \(j=0\), together with Equation [eq:source-18]. The fixed core ratio is positive, and hence \[a_n^{-2}\leq C\lambda^N\leq Cn \leq C_E\frac{n}{1+r}.\] This case includes repeated locations with either pair of signs. The deterministic Green factors in the two \(W_n\)’s are uniformly bounded on \(E\) by Lemma 25. They therefore preserve the bound. Increasing \(C_E\) handles the finitely many smaller lattice sizes.

Finally, the Fourier current expansion of the pinned XY law expresses each integer-charge correlation as a sum of nonnegative weights divided by a positive partition function. The fixed boundary imposes no neutrality constraint; repeated locations merely add their charges. Multiplication by the positive deterministic factors proves all the asserted nonnegativity statements. ◻

Proof. Let \(U\) be independent and uniform on the circle, and rotate every spin, including each boundary spin, by \(U\). Subtracting the common boundary angle recovers the original fixed-boundary law. Put \(X=\operatorname{Re}(e^{iU}W)\). The finite-graph XY Lee–Yang Theorem, with the common uniformly distributed boundary convention, says that the entire function \(\mathbb Ee^{zX}\) has only imaginary zeros [13]. One may obtain this boundary convention on simple graphs by adjoining a vertex of observable weight zero, joining it ferromagnetically to every boundary vertex, and taking all these added couplings to infinity. The limiting spins on the boundary equal the new spin, whose angle is uniform by rotation invariance. The bounded observable gives locally uniform convergence of the Laplace transforms; the zero-free half-planes persist under this limit.

The random variable \(X\) is symmetric and bounded. Hence its Lee–Yang product representation [13] is \[\mathbb Ee^{zX} =e^{Bz^2}\prod_j(1+z^2/y_j^2),\qquad B\geq0,\quad \sum_jy_j^{-2}<\infty,\quad B+\sum_jy_j^{-2}=\tfrac12\mathbb EX^2.\] The zeros are listed as pairs \(\pm i y_j\), with \(y_j>0\) and multiplicity. Since uniform rotation gives \(\mathbb EX^2=v/2\), the inequality \(1+u\leq e^u\) yields \[ \mathbb Ee^{tX}\leq e^{t^2v/4},\qquad t\in\mathbb R. \tag{26}\] The variable \(Y=\operatorname{Im}(e^{iU}W)\) has the same law as \(X\). If \(v>0\), Chernoff’s Inequality applied to both signs of \(X\) and \(Y\) therefore gives \[\mathbb P\{|W|>r\} \leq \mathbb P\{|X|>r/\sqrt2\}+\mathbb P\{|Y|>r/\sqrt2\} \leq4e^{-r^2/(2v)},\qquad r>0.\] Integrating this bound against \(p r^{p-1}\,dr\) gives \(\mathbb E|W|^p\leq2p(2v)^{p/2}\Gamma(p/2)\), which implies Equation (24). When \(v=0\), the assertion is immediate. The auxiliary rotation has left \(|W|\) unchanged.

For Equation (25), orient the edges arbitrarily and expand each interaction in its absolutely convergent Fourier series. If an edge has coupling \(b\geq0\), its coefficients are \[I_k(b)=\sum_{r=0}^\infty \frac{(b/2)^{2r+|k|}}{r!(r+|k|)!}\geq0, \qquad k\in\mathbb Z.\] At \(b=0\) these are interpreted as \(I_0(0)=1\) and \(I_k(0)=0\) for \(k\ne0\). Integrating each interior angle imposes the divergence constraint prescribed by the inserted charges on the integer edge currents. Boundary angles are zero, so they contribute a factor one and impose no divergence constraint. Each surviving summand is nonnegative, and the partition function in the denominator is positive. Repeated vertices simply add their integer charges. ◻

Proof. Fix a compact set \(E\Subset D\) containing the supports of the tests, and choose \(0<\rho<1\) so that its closed \(2\rho\)-neighborhood lies in \(D\). On that neighborhood write \[G_D(u,v)=\frac1{2\pi}\log\frac1{|u-v|}+R_D(u,v), \qquad q=\frac1{2\pi K}<2.\] The remainder \(R_D\) is smooth and bounded, including across the diagonal. Let \(C_{a,b}(x,y)\) be the covariance of the circle averages of \(\Phi_D/\sqrt K\) with centers \(x,y\in E\) and radii \(a,b\leq\rho\). We first establish, uniformly in both radii, \[ C_{a,b}(x,y) =q\log\frac1{\max\{|x-y|,a,b\}}+O_{E,K}(1). \tag{28}\] To see this, put \(r=|x-y|\) and \(M=\max\{r,a,b\}\). The double circle average of the logarithm of the distance equals \[L_{a,b}(r)=\frac1{2\pi}\int_0^{2\pi} \log\max\{a,|r+be^{it}|\}\,dt.\] Here we have used the elementary circle-mean identity \((2\pi)^{-1}\int_0^{2\pi}\log|z-ae^{it}|\,dt =\log\max\{|z|,a\}\). The integrand is at most \(\log(2M)\). If \(a=M\), it is at least \(\log M\). Otherwise, its integral is bounded below by the circle mean of \(\log|r+be^{it}|\), which is \(\log\max\{r,b\}=\log M\). Thus \(\log M\leq L_{a,b}(r)\leq\log M+\log2\). Together with the bounded remainder this proves Equation (28), even for coincident centers. The finite logarithmic integrals also justify the covariances of the circle averages.

For arbitrary radii \(\varepsilon_i\leq\rho\), Gaussian integration and cancellation of the full diagonal variances give \[ \mathbb E\prod_{i=1}^m V_{\varepsilon_i}^{\sigma_i}(x_i) =\exp\left\{-\sum_{i<j}\sigma_i\sigma_j C_{\varepsilon_i,\varepsilon_j}(x_i,x_j)\right\}. \tag{29}\] At the pointwise level \(V_\varepsilon^-\) denotes \(\overline{V_\varepsilon}\). To dominate this expression for all signs and all collision patterns, suppose that \(m\geq2\) and the centers are distinct, and set \[d_i=\min\{\rho,\min_{j\ne i}|x_i-x_j|\}, \qquad r_i=\max\{\varepsilon_i,d_i/4\}.\] All circles with radii \(r_i\) remain in \(D\). For \(i\ne j\), \[\max\{|x_i-x_j|,r_i,r_j\} =\max\{|x_i-x_j|,\varepsilon_i,\varepsilon_j\}.\] Consequently enlarging the radii changes each off-diagonal covariance by a bounded amount. If \(C^r\) is the covariance matrix for the enlarged circles, its positive semidefiniteness gives \[-\sum_{i<j}\sigma_i\sigma_j C^r_{ij} =\frac12\sum_i C^r_{ii} -\frac12\sum_{i,j}\sigma_i\sigma_j C^r_{ij} \leq\frac12\sum_i C^r_{ii}.\] Since \(C^r_{ii}\leq q\log(4/d_i)+C_{E,K}\), we obtain \[ 0<\mathbb E\prod_i V_{\varepsilon_i}^{\sigma_i}(x_i) \leq C_{E,K,m}\prod_i d_i^{-q/2}. \tag{30}\] This bound is independent of every radius, including their relative sizes. For \(m=1\) the expectation is exactly one.

We verify explicitly that the right side is integrable on \(E^m\). Put \(\alpha=q/2<1\) and use \[d_i^{-\alpha} \leq\rho^{-\alpha}+\sum_{j\ne i}|x_i-x_j|^{-\alpha}.\] Expanding the product produces finitely many directed graphs on \(\{1,\ldots,m\}\), with at most one outgoing edge from each vertex and no self-edge. An edge \(i\to j\) contributes \(|x_i-x_j|^{-\alpha}\). Each connected component is a tree, or consists of one directed cycle with trees attached. Vertices with no incoming edge can be integrated first: each has at most one remaining incident edge, and \[ \sup_{y\in E}\int_E |x-y|^{-s}\,dx<\infty,\qquad 0\leq s<2. \tag{31}\] This follows by polar integration over a fixed disk containing \(E-E\). It removes all trees at a uniform cost. A cycle on two vertices contributes \(|x-y|^{-2\alpha}\) and is integrable because \(2\alpha=q<2\). For a cycle on at least three vertices, integrate one vertex using Cauchy–Schwarz: \[\int_E |x-y|^{-\alpha}|x-z|^{-\alpha}\,dx \leq \left(\int_E|x-y|^{-2\alpha}\,dx\right)^{1/2} \left(\int_E|x-z|^{-2\alpha}\,dx\right)^{1/2} \leq C_{E,\alpha}.\] The remaining graph is a path, which can be integrated from its ends. Isolated vertices contribute \(|E|\). Hence every term of the expansion is integrable, including near simultaneous collisions of any number of centers.

At distinct centers, \(C_{\varepsilon_i,\varepsilon_j}(x_i,x_j)\to G_D(x_i,x_j)/K\) as all radii tend to zero, with no restriction on their relative rates. Equations (29) and (30) now give convergence of all integrated mixed kernels by dominated convergence. In particular, the second mixed kernels with radii \(\varepsilon,\delta\) show that \(V_\varepsilon(f)\) is Cauchy in \(L^2\). Taking equal numbers of positive and negative signs gives uniform bounds for every even absolute moment. For any finite \(p\), choose an even order greater than \(p\). These bounds and \(L^2\) convergence give convergence in \(L^p\) by uniform integrability. Hölder’s Inequality then passes each mixed product to the limit and proves Equation (27).

It remains to realize these limits as one random distribution. Choose a flat torus whose fundamental square contains \(\overline D\) in its interior, with normalized Fourier basis \(e_\nu\), \(\nu\in\mathbb Z^2\). For a fixed \(\chi\in C_c^\infty(D)\), Equation (30) with two factors implies \[\sup_{\nu,\ 0<\varepsilon<\varepsilon_\chi} \mathbb E|V_\varepsilon(\chi e_\nu)|^2<\infty\] for sufficiently small \(\varepsilon_\chi>0\). Each Fourier coefficient is Cauchy in \(L^2\), and \(\sum_{\nu\in\mathbb Z^2}(1+|\nu|^2)^{-2}<\infty\). Dominated convergence in this series proves that \(\chi V_\varepsilon\) is Cauchy in \(L^2(\Omega;H^{-2})\) on the torus. Choose a countable exhaustion by smooth cutoffs which equal one near the supports of their predecessors. Their limits are compatible, since multiplication by a smooth function is continuous on \(H^{-2}\). They define an almost surely \(H^{-2}_{\mathrm{loc}}\) distribution \(V_{K,D}\), and testing recovers the scalar limits already constructed. Localization identifies these torus Sobolev spaces with the Euclidean local spaces on \(D\). This proves all the asserted convergence. In particular the normalization uses the complete Dirichlet variance, including its finite spatial part. ◻

Proof. Write \(h=n^{-1}\). For a fixed \(x\in B\cap D_n^\circ\), the shell \(jh\leq|x-y|<(j+1)h\) contains at most \(C(j+1)\) lattice points. Equation [eq:source-20] thus implies \[h^2\sum_{y\in B\cap D_n^\circ}(|x-y|+h)^{-1} \leq Ch\sum_{0\leq j\leq C(d/h+1)}1 \leq C(d+h).\] There are at most \(C(d/h+1)^2\) choices of \(x\), so a second summation, including its factor \(h^2\), proves the first bound in Equation (33). Summation over a fixed compact set gives a uniform bound for its indicator. Positivity of the opposite-sign two-point expectation then gives, for arbitrary complex \(f\) supported there, \[\mathbb E_n^0|S_n^+(f)|^2 \leq h^4\sum_{x,y\in E\cap D_n^\circ} |f(x)|\,|f(y)|\, \mathbb E_n^0[W_n^+(x)W_n^-(y)] \leq C_E\|f\|_\infty^2.\]

For a nonnegative weight \(w\), the coefficients of the spins in \(S_n^+(w)\) are \[\lambda_x=n^{-2}w(x)a_n^{-1} \exp\left\{\frac{G_n(x,x)-G_n(0,0)}{2K_*}\right\}\geq0.\] Lemma 29 applies to these exact coefficients for each \(n\). It proves the second box bound from the first. For a complex weight, decompose \[f=(\operatorname{Re}f)_+-(\operatorname{Re}f)_- +i(\operatorname{Im}f)_+-i(\operatorname{Im}f)_-.\] Each nonnegative summand is bounded by \(\|f\|_\infty\) and supported in \(E\). Minkowski’s Inequality and the Lee–Yang bound prove the second estimate in Equation (32). Conjugation proves the same bounds for the minus field. ◻

Proof. Let \(E\Subset D\) contain all supports. The case \(m=1\) follows from Equation [eq:source-17], uniformly on \(E\), and Riemann summation. Assume \(m\geq2\) and expand the moment as a sum over \((x_1,\ldots,x_m)\in(D_n^\circ)^m\). All signed correlations in this sum are nonnegative by Equation (25) and the positivity of the deterministic normalizers.

Fix two indices \(i\ne j\) and a small \(d>0\). Enlarging the squares of a grid of side \(d\) by a fixed factor gives a family \(\mathcal B_d\) of at most \(C_Ed^{-2}\) squares, each of side at most \(C d\), such that every pair \(x_i,x_j\in E\) with \(|x_i-x_j|<d\) lies in one of these squares. All these squares lie in a fixed larger compact set \(E'\Subset D\) when \(d\) is small. Positivity allows us to bound the absolute value of the restricted moment sum by \[\left(\prod_{\ell=1}^m\|f_\ell\|_\infty\right) \sum_{B\in\mathcal B_d} \mathbb E_n^0\left[ S_n^{\sigma_i}(1_B)S_n^{\sigma_j}(1_B) \prod_{\ell\ne i,j}S_n^{\sigma_\ell}(1_{E'}) \right].\] Each expectation is nonnegative. Its absolute value is at most the expectation of the product of absolute values. Hölder’s Inequality with \(m\) equal exponents and Lemmas 29 and 31 bound it by \[C_{E',m} \|S_n^+(1_B)\|_{L^2}^2 \leq C_{E',m}(d+n^{-1})^3.\] Indeed the two box factors each contribute their \(L^2\) norm times a constant depending only on \(m\), and every other factor has a uniformly bounded \(L^m\) norm. Summing over all pairs proves \[ \limsup_{n\to\infty} n^{-2m}\!\!\sum_{\substack{x_1,\ldots,x_m\in D_n^\circ\\ \min_{i<j}|x_i-x_j|<d}} \left(\prod_i|f_i(x_i)|\right) \mathbb E_n^0\prod_i W_n^{\sigma_i}(x_i) \leq C_{E,m,f_1,\ldots,f_m}d. \tag{35}\] In particular repeated lattice sites, of every multiplicity and every choice of signs, are included in this estimate.

Choose a continuous cutoff on configurations which vanishes when some pair is at distance at most \(d\) and equals one when every pair is at distance at least \(2d\). On its support, the uniform separated-point convergence in Equation [eq:source-17] turns the lattice moment sum into the Riemann sum of the continuum kernel. The cutoff makes the product with the test functions continuous across the excluded diagonals. For fixed \(d\) these sums therefore converge to their continuum integrals. Equation (35) bounds the omitted lattice contribution by a quantity tending to zero with \(d\), after taking the upper limit in \(n\). The omitted continuum contribution tends to zero by the integrable domination in Equation (30). Letting \(d\downarrow0\) proves Equation (34). ◻

Proof. Fix a real linear combination \(Z_n\) of the real and imaginary parts of the \(k\) lattice evaluations, and let \(Z\) be the same linear combination of the continuum evaluations. Equation (32) gives \(\|Z_n\|_{L^p}\leq C\sqrt p\) for all \(p\geq2\), with \(C\) depending on the tests and the chosen linear combination. Consequently, for every \(t\geq0\), \[ \sup_n\mathbb E_n^0 e^{t|Z_n|}<\infty. \tag{36}\] Indeed the exponential series is dominated, apart from its first two terms, by \(\sum_{p\geq2}(Ct\sqrt p)^p/p!<\infty\). Proposition 32 gives convergence of the even moments of \(Z_n\) to those of \(Z\). Hence \(Z\) satisfies the same bound at even orders, and at all remaining orders by Hölder’s Inequality. It too has every exponential moment. Absolute convergence of the exponential series thus determines its characteristic function from its moments.

The finite-dimensional lattice vectors are tight by their second-moment bounds. Along a weakly convergent subsequence, the higher uniform moment bounds make every polynomial in their real and imaginary coordinates uniformly integrable. The limit therefore has the moments given in Proposition 32. Each of its real projections has the same even-moment bounds as \(Z\), hence is exponentially integrable and has the same characteristic function as \(Z\). The Cramér–Wold Criterion identifies the vector law. Every convergent subsequence has this law, which proves convergence along the full sequence. For the stated mixed moments, use the sign convention \(\overline{S_n(f)}=S_n^-(\overline f)\) in Proposition 32. ◻

Proof. Use the larger flat torus from the continuum construction and its normalized Fourier basis \(e_\nu\), \(\nu\in\mathbb Z^2\). Choose real smooth cutoffs \(\chi_j\) with compact support in \(D\), equal to one on an increasing exhaustion of \(D\), and with \(\chi_{j+1}=1\) near \(\mathop{\mathrm{supp}}\chi_j\). Since the Fourier functions have uniformly bounded suprema, Equation (32) gives \[\sup_{n,\nu}\mathbb E_n^0|S_n(\chi_j e_\nu)|^2\leq C_j.\] Fourier summation yields \[ \sup_n\mathbb E_n^0\|\chi_j S_n\|_{H^{-2}}^2 \leq C'_j\sum_{\nu\in\mathbb Z^2}(1+|\nu|^2)^{-2} <\infty. \tag{37}\] The embedding \(H^{-2}\hookrightarrow H^{-3}\) on this torus is compact. Indeed finite Fourier truncations have finite rank, and on their complements the ratio of the squared Sobolev weights is \((1+|\nu|^2)^{-1}\to0\).

Given \(\eta>0\), choose \(R_j<\infty\) so that the sum of the Markov bounds in Equation (37), with thresholds \(R_j^2\), is less than \(\eta\). With probability at least \(1-\eta\), all inequalities \(\|\chi_jS_n\|_{H^{-2}}\leq R_j\) hold simultaneously. The set of distributions satisfying these inequalities has relatively compact closure in \(H^{-3}_{\mathrm{loc}}(D)\): compact embedding gives a convergent subsequence for each localized field, and a diagonal choice gives convergence for all \(j\). The identities \(\chi_j(\chi_{j+1}S_n)=\chi_jS_n\) pass to the limits because smooth multiplication is continuous on \(H^{-3}\). Thus the limits define one distribution on \(D\). This proves tightness.

For completeness, the topology just used is the Euclidean local topology in the statement of Theorem 1. On distributions supported in any fixed compact subset of \(D\), the torus Sobolev norms and the corresponding Euclidean Sobolev norms are equivalent by localization inside its fundamental square. Moreover every compactly supported cutoff is supported where some \(\chi_j\) equals one. The countable family of these localized norms therefore generates exactly \(H^{-3}_{\mathrm{loc}}(D)\).

Every subsequential limit in this space has the test-function laws in Proposition 33, since evaluation on a smooth compactly supported test is continuous. These laws determine the distribution-valued law. For example, the countably many tests \(\chi_j e_\nu\) determine every localized Fourier coefficient; their real and imaginary parts generate the Borel sigma-field of the compatible subspace of the countable product of the separable spaces \(H^{-3}\). Lemma 30 places \(V_*\) in this space, so all subsequential limits have its law. Tightness and this uniqueness prove full-sequence convergence. ◻