Context and method

Nienhuis’s analysis of the dilute \(O(n)\) model predicted the honeycomb critical point and critical exponents [14]. Duminil-Copin and Smirnov proved the exact connective constant \(\sqrt{2+\sqrt2}\) by a parafermionic observable [4]. Their convention also uses mid-edge endpoints and counts visited vertices. We use the corresponding critical activity and give the local flux calculation with our boundary-port convention.

For critical strip-crossing mass, Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved convergence to zero [1]. Glazman and Manolescu obtained a logarithmic bound along a subsequence [9], and Krachun and Panagiotis proved a polynomial upper bound with exponent \(10^{-10}\) [13]. The predicted decay is \(h^{-1/4}\), as recorded in [9]. Theorem 1 gives this decay with fixed multiplicative constants, together with first-moment and positive-mass length estimates under the same finite bridge measure.

Yang–Baxter deformations provide a second context: Glazman and Manolescu proved invariance of the boundary two-point function for columnwise rhombic half-planes with Yang–Baxter weights and angles in \([\pi/3,2\pi/3]\) [9]. The dilute transfer-matrix setting also underlies the bulk spectral calculations of Zhou and Batchelor [17]. Polynomial exchange and fusion methods for dense and open dilute \(O(1)\) models [3, 7, 8] provide precedents for our finite construction; the periodic \(n=0\) identities and their physical contractions are proved here.

For systems of polygons separating two fixed points on a cylinder, we introduce a generating function with a variable factor per polygon. Polynomial exchange and fusion express it as a finite Laurent polynomial divided by a vacuum normalization. We estimate the numerator by a current expansion based on the classical level-one boson construction of Frenkel–Kac and Segal [6, 15, 5]; the thermal trace estimates and contour deformations are established here.

The denominator may be small, and an unequal marked interval leads to an open product of operators rather than a periodic trace. Our argument treats these two issues separately. At zero separating-loop weight the physical partition is exactly one. This special value supplies a saturation point for the analytic estimate and eventually a bounded-factor normalization. For the unequal marked intervals we construct an operator for the ordered particle expansion, prove compactness with quantitative state bounds, and control products on the two long intervals. A bound for closed periodic traces is used only after the operator realizing the open products has been constructed.

The geometry has an analogous normalization issue. Joining each chord to an exterior connector would appear to lose a critical bridge factor at every stage. We vary the intermediate cuts and apply a second moment estimate. The difference between two cut choices produces additional bridges whose averaged costs recover those apparent losses. A further gain near corners makes the sum over endpoint bins finite. Together with (5), this gives a second-moment bound at the required scale for the chord length and then the long-bridge mass in (3).

Proof route

Sections 2 and 3 establish the strip mass independently, first through a finite double-row interpolation formula and then through positive interpolation estimates. Section 4 constructs the periodic vacuum vectors, which represent the empty-connectivity eigenvalue, and identifies their physical polygon pairing. Section 5 transforms that finite polynomial into a current expansion, an algebraic reorganization of its particle sum, and bounds it, including all zero modes and contour changes. Section 6 obtains the preliminary saturation needed to identify a nonzero physical normalization.

Section 7 proves the boundary crossing and cylinder-barrier estimates, and converts positive nesting into a free-boundary length sum. Section 8 then sharpens the normalization to bounded factors. Section 9 inserts prescribed-edge observables, called magnetic insertions in the transfer calculation, and proves the unequal marked estimate. Finally, Section 10 closes and extends chords in the exterior, sums over marked pairs, and proves Theorem 1. Each stage keeps the physical length convention above; no change to a length-conditioned or infinite-path law occurs in the argument.

Local matrices and fusion

Let the three basis symbols be \(+,0,-\) identified with spins \(1,0,-1\). Write \(s=\exp(2iL)\). For spectral parameter \(u\) define normalized coefficients \[\begin{aligned} (a,h,v,c,d)(u)&=\frac{1}{Q(u)} \big(S\sin(b-u),\ S\sin u,\ \sin(b-u)\sin u,\;\\ &\hspace{3em}\sin(b-u)\sin(2L-u),\ \sin u\sin(u-L)\big),\\ Q(u)&=\sin(2L+u)\sin(b+u). \end{aligned}\] The matrix \(R(u)\) on an ordered pair of spin spaces acts by \(c\) on \(++\) and \(--\); on each of \(\operatorname{span}(+0,0+)\) and \(\operatorname{span}(-0,0-)\) by \(\left(\begin{smallmatrix}v&a\\a&v\end{smallmatrix}\right)\); and on \((+-,00,-+)\) by \[\begin{pmatrix} d&h/s&c+d/s^2\\ h s&1&h/s\\ c+d s^2&h s&d \end{pmatrix}.\] Use boundary variables \(e\) with \(e^2=0\), commuting. Define \[\begin{gathered} k(t)=\frac{\sin(\gamma-t)}{\sin(\gamma+t)},\quad l(t)=\frac{\sin(2t)}{\sin(\gamma+t)},\\ D_e(t)=\operatorname{diag}(k,1,k)+ e\,l(E_{+,0}+s E_{0,-}),\\ K_e(t)_{p,-m}=s^m (D_e(t))_{p m}, \end{gathered}\] where \(E_{ij}\) are matrix units and \(K_e\) is used as a two-spin vector. On a row of \(N\) (site) spins with auxiliary spin \(\alpha\), rapidities \(\xi_i\), put \[U_\alpha(t)=R_{\alpha 1}(t+\xi_1)\cdots R_{\alpha N}(t+\xi_N).\] Subscripts indicate the ordered spaces acted upon; other factors are spectators. Contract two auxiliary rows in the double row operator (\(r=b-t\)) \[T(t) = K_{e_\ell}(r)_{AB}^{T} U_A(t) U_B(r) K_{e_r}(t)_{AB}.\] Here \(e_\ell^2=e_r^2=0\) separately, and superscript \(T\) is ordinary transpose. The nilpotent variables retain at most one boundary insertion on each side. Figure 1 shows the order of the two auxiliary rows and the site spaces.

We record local identities and their algebraic verification. Equalities are meromorphic (used first away from singularities). Let \(P\) swap a pair of spin spaces, \(F\) flip spins in a space, \(J=\operatorname{diag}(-1,1,-1)\). Define \[\begin{gathered} \Omega=\sum_p s^{-p}|p,-p\rangle,\qquad W|\pm\rangle=\kappa(|\pm,0\rangle+|0,\pm\rangle),\\ W|0\rangle=|00\rangle+\kappa\sum_{p=\pm1}s^{-p}|p,-p\rangle,\quad Z=W^*=\bar W^T. \end{gathered}\] The image slots of \(W\) will be indicated by a pair of labels (ordered); its input slot can then be regarded as a fused spin. We have \[ \begin{gathered} R_{21}(u)=R_{12}^T(u),\qquad R_{12}(u)R_{12}^T(-u)=I,\\ R(u+\pi)=J_1 R(u)J_1,\qquad R(b-u)_{ij,kl}=s^{k-i}R(u)_{-i,l;-k,j},\\ R(0)=P,\qquad R(b)=\Omega\bar\Omega^T,\qquad R(2L)=W Z. \end{gathered} \tag{7}\] Indeed \(Q(b-u)=Q(u)\), swapping \(a,h\) and \(c,d\) at reflected argument, which gives crossing by the displayed entries. Unitarity on the \(\pm2,\pm1\) blocks uses \(c(u)c(-u)=1\), \(v(u)/a(u)=\sin u/S\) and \(a(u)a(-u)=S^2/(S^2-\sin^2 u)\). On the zero block use additionally (prime denotes argument \(-u\) here) \(h+c h'=0\), \(h h'+c d'+c' d=0\), by the sine products, and \(s^2+s^{-2}=0\). Values in (7) follow by substitution. Consequently \[R_{23}(u-b)R_{13}(u)\Omega_{12}=\Omega_{12} I_3,\] where \(I_3\) denotes the identity on slot 3 (by crossing, the output \((p,q,k)\) at input \(j\) is \(s^{-p}(R(u-b)R(b-u)^T)_{qk;-p,j}\)).

The basic fusion identity and its adjoint are \[ R_{23}(u-L)R_{13}(u+L)W_{12}=W_{12} R_{f3}(u),\qquad Z_{12} R_{13}(u+L)R_{23}(u-L)=R_{f3}(u) Z_{12}, \tag{8}\] where \(f\) labels the fused space. Here and below adjoints can be taken at nonsingular real parameters first (\(R\) Hermitian). We give details of the elementary check to fix normalizations. Multiply the first formula by \(R_{32}(L-u)\) using (7), and write a subscript \(+\) for argument \(u+L\), primes for \(L-u\), no decorations for \(u\). Expanding on inputs \((f,3)\) gives the following sufficient equalities (spin-flipped inputs conjugate the same ones): \[\begin{array}{l|l} ++&cv'=c_+,\ cc'=v_+,\ ca'=a_+\\ +0&v+a h'=v_+,\ vv'+aa'/\kappa=1,\ va'+av'/\kappa=a_+,\;\\ &v h'+a d'=0,\ ac'=h_+\\ 0+&\kappa(a+v h')=a_+,\ av'+va'/\kappa=h_+,\ \kappa aa'+vv'=v_+,\;\\ &a h'+v d'=c_+,\ vc'=d_+\\ 00&\kappa(a'+v'h)=h_+,\ v'+a'h=v_+,\ h'+\kappa h c'=\kappa a_+\\ +-&h+\kappa c h'=\kappa h_+,\ hh'+\kappa(c d'+c'd)=0,\\ &d v'+h a'=d_+,\;d a'+h v'=0,\text{ and the }++\text{ identities}. \end{array}\] To verify, put \(t_j=\sin(u+jL)\); the numerator rows and denominators are \[\begin{array}{c|rrrrr|l} &a&h&v&c&d&Q\\ &-S t_{-3}&S t_0&-t_{-3}t_0&t_{-3}t_{-2}&t_0t_{-1}&t_2t_3\\ '&S t_2&-S t_{-1}&-t_{-1}t_2&t_2t_1&t_{-1}t_0&-t_{-3}t_4\\ +&-S t_{-2}&S t_1&-t_1t_{-2}&t_{-2}t_{-1}&t_1t_0&t_3t_4 . \end{array}\] Thus \(QQ'/Q_+=-t_{-3}t_2\). The \(++\) row and \(v h'+a d'=0\), \(ac'=h_+\), \(vc'=d_+\) follow immediately; for the other identities in rows \(+0,0+\) use \(t_{-2}t_2=t_0^2-S^2\), \(t_{j-1}+t_{j+1}=t_j/\kappa\), \(t_{-3}t_4=t_0t_1-S^2/\kappa\), \(t_3t_4=S^2/\kappa-t_0t_{-1}\), \(t_0t_{-1}-t_{-2}t_1=\kappa S^2\). For the last two rows one also has \(t_2t_1-t_{-2}t_{-1}=t_0t_4/\kappa\), \(t_{-3}t_{-2}+t_2t_1=S^2/\kappa\), and can interchange primed and unprimed quantities (swaps \(h_+,a_+\) and \(d_+,c_+\)).

These imply Yang–Baxter: \[R_{12}(u-v) R_{13}(u) R_{23}(v)=R_{23}(v) R_{13}(u) R_{12}(u-v).\] Indeed at \(u-v=0\) use the permutation; at \(u-v=2L\) use \(R(2L)=W Z\) and (8); at \(u-v=b\) use (7) and the analogous rank-one identity shown after (7), and its adjoint. At \(u=0,2L\) reduce to the first two cases by multiplying on both sides by \(R_{23}(v)^{-1}=R_{32}(-v)\) and permuting the labels (new order \(1,3,2\)). For generic fixed \(v\) these give five values modulo \(\pi\); clearing denominators, each entry has trigonometric degree \(\le4\) in \(u\) and is either periodic or antiperiodic by \(\pi\) by (7), so vanishes identically.

Commuting and fusing double rows

We now derive identities for \(T\) by row exchange. Yang–Baxter gives \(R_{ij}(u-v) U_i(u)U_j(v)=U_j(v)U_i(u) R_{ij}(u-v)\). In particular swapping the two rows of \(T\) gives \(T(t)=T(r)\): the input swap uses \(R(t-r)K_e(t)=P K_e(r)\) and the output swap its \(r,t\) version via unitarity. Also \(T(t+\pi)=T(t)\) since \(K_e(t+\pi)=J_1 J_2 K_e(t)\).

For \(T(t)T(y)\), take labels \(A,B,C,D'\) of parameters \(t,r,y,b-y\) respectively (below write \(D'\) for a label). To interchange the row pairs, exchange \(BC,AC,BD',AD'\) in that order, acting on the right boundary vector \(V=K_e(t)_{AB}K_e(y)_{CD'}\) (\(e=e_r\)) by the corresponding \(R\) product. Invariance of the vector reduces by (7) to \[R_{AC}(t-y)R_{BC}(b-t-y)V =R_{BD'}^T(t-y)R_{AD'}^T(b-t-y)V .\] This is (10) by crossing. Indeed identify a vector \(X\) with the matrix with entries \(s^{-m-n}X_{i,-m,j,-n}\) in row \(i,j\) and column \(m,n\). Under this identification \(R_{BC}(b-t-y)V\) is \(D_{e,1}(t) R(t+y)^T D_{e,2}(y)\) and \(R_{AD'}^T(b-t-y)V\) is \(D_{e,2}(y) R(t+y) D_{e,1}(t)\). The remaining \(R\)’s act on the left and right respectively as in (10). On the left boundary (covector written as a transposed column), the inverse-transposed exchanges appear, exactly the analogous computation with \(t,r\) exchanged and \(y,b-y\) exchanged, \(e=e_\ell\). Thus all \(T\) commute. Further \(T(0)=T(4L)=I\): at \(0\) the input is \(K_e(0)=\bar\Omega=P\Omega\) and the two rows act trivially on it by rank-one crossing after (7). The output pairing with this vector is 1. At \(4L\), shift the second parameter \(r\) by \(\pi\) (conjugate that slot by \(J\)); the input becomes again \(\bar\Omega\), the parameter difference again \(b\) (higher parameter second), and the output pairing is 1.

For a product \(T(t-L)T(t+L)\) use labels \(B,C,A,D'\) with parameters \(t-L,r+L,t+L,r-L\) in that order. Swap middle rows by \(R_{CA}(r-t)\). We obtain product order \(B A C D'\) between boundary column vectors \(G_r,G_\ell\) (output transposed): \[\begin{split} G_r &= R_{CA}(b-2t) K_{e_r}(t-L)_{BC} K_{e_r}(t+L)_{AD'},\\ G_\ell &= R_{CA}(b-2r) K_{e_\ell}(r+L)_{BC} K_{e_\ell}(r-L)_{AD'} . \end{split}\] With fused slots \(f,g'\) for \(AB,C D'\) respectively, (9) gives \[(1\otimes Z_{C D'})G_r=(W_{AB}\otimes1)K_{e_r}(t)_{f g'},\qquad (W_{AB}^T\otimes1)G_\ell=(1\otimes Z_{C D'}^T)K_{e_\ell}(r)_{f g'} .\] Indeed the matrix given by \(s^{-m-n} (G_r)_{p q;-m,-n}\) (slots indexed \(AB;C D'\), columns ordered \(n,m\)) by crossing is \(D_{e_r,2}(t-L) R(2t) D_{e_r,1}(t+L)\), and similarly for \(G_\ell\) with \(r,+L,-L\). Also \(Z_{-j;-m,-n}=\bar W_{n,m;j}\) and it is zero unless \(j=m+n\).

By (8), \(U_B U_A\) preserves the \(W_{AB}\) subspace acting by \(U_f(t)\), and \(U_C U_{D'}\) preserves \(\ker Z_{C D'}\) acting on the quotient by \(U_{g'}(r)\). Moreover:

• At \(t=L-\xi_i\) the local product \(R_{Bi}(0) R_{Ai}(2L)\) has range in \(W_{AB}\), hence so does \(U_B U_A\).

• At \(t=L+\xi_i\) the local product \(R_{Ci}(b) R_{D'i}(L)\) kills \(\ker Z_{C D'}\), hence so does \(U_C U_{D'}\). Indeed its adjoint has range in the image of \(W_{C D'}\): \(R_{D'i}(L)\) maps \(\Omega_{Ci}\) into this image by crossing at \(u=2L\), \(R(2L)=WZ\) (output \(c,d,j\) for input \(v\) on \(D'\) has coefficient \(s^{-c}R(L)_{dj;v,-c}=s^j R(2L)_{cd;-j,v}\)).

Consequently at either of those values the contracted product reduces to \(T(t)\). Indeed the part of \(G_r\) in arbitrary slots \(AB\) tensor \(\ker Z_{CD'}\) is then harmless (either killed, or mapped to \(W_{AB}\otimes\ker Z_{CD'}\) annihilated by the output); the rest can be lifted through \(W_{AB}\) from \(K_{e_r}(t)_{f g'}\), and (8) and the formula for \(G_\ell\) give precisely \(T(t)\). We have shown, first for generic rapidities, \[ T(t-L)T(t+L)=T(t)\quad\text{at }t=L\pm\xi_i. \tag{11}\]

The scalar crossing function

The row identities now have to be connected to the positive path sum. The next statement collects the scalar data that will enter the finite functional.

The finite functional

A positive determinant formula

Retain the measure and spaces in (6), and put \[\begin{gathered} c_1=C_{L/2},\quad c_3=C_{3L/2},\quad h_0=1/D,\quad h_1=c_1/D,\\ F_0=M+\mathbf R c_1,\quad F=F_0+\mathbf R c_3,\quad G=M+\mathbf R g \end{gathered}\] (using real spans in the following). Here \(N\) remains the strip width, while \(d=\dim F_0=2N+1\) is the number of interpolation nodes. A complete jet at a label of multiplicity \(k\) consists of derivatives of orders \(0,\ldots,k-1\). Use jets of \(C_{\sqrt\beta}(x)\) in \(\beta\) as bases, ordered first by \(\beta\), then by derivative order. Thus \(M\) takes labels \(0,L^2\) each with multiplicity \(N\); the \(g\) basis row in \(G\) is inserted between. For \(X=(x_1<\dots<x_d)\) in \((0,\infty)\), let \(\nu\) denote the probability proportional to \[\det F_0(X)\det G(X)\prod_i d\mu(x_i)\] (evaluation matrix with functions as rows). Indeed \(g=\int_0^L C_t\,dt\), so the \(G\) determinant integrates that for \(G_t=M+\mathbf R C_t\), \(0<t<L\).

Write \(e_y H=H(y)-I_{F_0,X}H(y)\) for evaluation minus interpolation at \(X\) using \(F_0\). Also use \(E_y H=H(y)-I_{F,(0,X)}H(y)\). The determinant integrations below are instances of Andréief’s identity; see [16]. We retain the interpolation calculation to fix its oblique projection and evaluation columns. Gram/Cramer identities using (6) give, for \(s_*=\rho/(4a_*)\) there, \[\mathcal L(H)/(4a_*)=\nu[e_{-4}H]+s_*\nu[e_0 H].\] Indeed integrating a determinant product gives the determinant of the pairings by multilinearity, so the mean interpolant is the unique linear projection onto \(F_0\) preserving the moments against \(G\). Also \[\nu[e_0 c_3]\,\nu[e_{-4}h_0]-\nu[e_{-4}c_3]\,\nu[e_0 h_0] =\nu[(e_0 c_3) E_{-4}h_0].\] For this append rows \(c_3,h_0\) and evaluation columns \(0,-4\) to the \(F_0\) determinant and integrate against \(\det G(X)\) (Schur complement). Since \(d\) is odd, \(e_0 c_3<0\) by ordered jet positivity. Let \(\nu_3\) be the tilt of \(\nu\) by \(-e_0 c_3\). It will suffice to show \[ \begin{gathered} \nu[-e_0 c_3]\gtrsim N^{-1/2},\qquad |\nu[e_0 h_i]|\le N^{-1+o(1)},\qquad |\nu[e_{-4}h_i]|=O(1)\quad (i=0,1),\\ \nu_3[E_{-4} h_0]\asymp N^{-1/4}. \end{gathered} \tag{12}\] For the elimination, abbreviate \[A_y=\nu[e_yc_3],\qquad H_{j,y}=\nu[e_yh_j] \quad(y=0,-4;\ j=0,1).\] The condition \(\mathcal L(c_3-h_1)=0\) gives \[s_*=-\frac{A_{-4}-H_{1,-4}}{A_0-H_{1,0}},\] whenever the denominator is nonzero. The determinant identity above, together with the definition of \(\nu_3\), then gives the exact formula \[\frac{\mathcal L(h_0)}{4a_*} =\frac{A_0\nu_3[E_{-4}h_0] +H_{1,-4}H_{0,0}-H_{1,0}H_{0,-4}} {A_0-H_{1,0}}.\] The estimates in (12) imply that \(|A_0|\gtrsim N^{-1/2}\) and \(|H_{1,0}|\le N^{-1+o(1)}\). Hence the denominator is nonzero for large \(N\), the coefficient of the positive residual is \(1+O(N^{-1/2+o(1)})\), and the remaining quotient is \(O(N^{-1/2+o(1)})\). The estimate for \(\nu_3[E_{-4}h_0]\) therefore proves \(B_N\asymp N^{-1/4}\).

Characteristic products

We isolate the elementary identity used in its proof, so that the polynomial comparison has an explicit starting point.

In particular under either \(\eta\) the probability of having \(x_{N-1}<N^{3/2}\) is exponentially small (the growing-parameter product there is \(\ge (3/2)^{N-1}\)).

Interpolation residuals

The characteristic products will control the errors at \(0\) and \(-4\) once the following pointwise comparison is established.

Under the measures needed in (12), the omitted factors in the upper-product truncation of (15) have \(v_i\ge N^{3/2}\) on the event \(x_{N-1}\ge N^{3/2}\). There are \(O(N)\) such factors, so the ratio between the truncated and full products is at most \(\exp(O(N^{-1/2}))\). The complementary event has exponentially small probability by Lemma 6, and the pointwise upper bound is uniformly bounded there. For \(a=0\) we can bound above with \(a\) replaced by a small positive constant in these products, then send it to zero in the exponent estimate. Applying (13) therefore proves all the \(h_i\) estimates of (12) (\(b\lambda_*/\pi=1\), \(b(\lambda_*-2)/\pi=1/4\)).

The normalizing coefficient

Finally consider \(O(X)=-e_0 c_3>0\). This is increasing in the ordered nodes by jet positivity: when just one node of rank \(i\) (starting at 0) is increased without passing its neighbor, the change of \(e_0 c_3\) equals the old interpolation weight at 0 of that node (sign \((-1)^i\)) times the new error on \(c_3\) at the old node (sign \((-1)^{d+i}\)). Fix any \(G_t\) as before in place of \(G\). The expectation of \(O\) with base weight \(\mu\) is bounded below by that with \(\mu_b\) of (14). Indeed the density ratio of the base weights is increasing: up to constants it is \[\frac{z(\cosh(4z)+\cosh(2z)-\cosh(3z))}{\sinh(4z)},\qquad z=L\sqrt{x},\] with increasing numerator/denominator series coefficient ratios \(\frac{2j+1}{4}(1+(1/4)^j-(9/16)^j)\).

Under \(\mu_b\) the mean interpolant at 0 on \(A_m\) of (14) is rational in \(w=w_m\) with denominator \(Q_B(w)=\prod(1-w b_i')\), numerator degree \(<d\) (\(b_i'\) are \(w\)-labels of \(G_t\) with multiplicities). This follows by preserving the pairings against \(G_t\) and (14), taking jets. At the labels \(a_i'\) of \(F_0\) it matches \(A_m(0)=1/\sqrt{1+w}\) with multiplicities. These facts still compute the mean residual at \(w=1\), \(m=3L/2\) by cross-integrability. Use the mixture of \(1/(s+w)\), \(s>1\), with density \(1/(\pi\sqrt{s-1})\) for \(1/\sqrt{1+w}\). Rational interpolation gives the absolute residual for each integrand at \(w=1\) (all with sign \((-1)^d\)) \[\frac{1}{1+s}\prod_i\frac{1+s b_i'}{1-b_i'}\prod_i\frac{1-a_i'}{s+a_i'}.\] Since all labels are in \([0,1/3]\), identical between the multisets except for one, this is bounded below uniformly on \(1<s<1+1/d\). Thus the mean \(O(X)\gtrsim d^{-1/2}\), transferring to \(\mu\) by the comparison and then to \(G\) by positive mixture. This completes (12). Substitution into the finite functional of Theorem 4 gives \(B_N\asymp N^{-1/4}\), the strip-mass assertion of Theorem 1.

Periodic eigenlines

Write \(K(\xi)_{pp}=\xi^p\) for the diagonal spin twist, \(p=+,0,-\). In this calculation transfer is single-row, on site spins, and we use the sector of total spin zero: \[T_\xi(t)=\operatorname{Tr}_a K_a(\xi)\,R_{a1}(t+v_1)\cdots R_{aN}(t+v_N),\qquad \xi=i\ \text{or }-i.\] We use the empty site vector (all zeros), denoted by index \(\varnothing\). Site order is as displayed even though operator action is from the right. By trigonometric degree at most \(D\) we mean Laurent exponents between \(-D\) and \(D\) in \(e^{iv_j}\). If the parity under \(v_j\mapsto v_j+\pi\) is \((-1)^D\), multiplication by \(e^{iDv_j}\) gives a polynomial of degree at most \(D\) in \(e^{2iv_j}\); hence \(D+1\) distinct zeros modulo \(\pi\) force it to vanish.

Interpolation of the two-point function

Criticality with the boundary-port convention.

We adapt the Hammersley–Welsh bridge decomposition, as used in [4], to the present strip and endpoint convention. In particular \(\kappa\) really is the critical single-walk weight (radius of convergence of the usual length series from a fixed honeycomb vertex). Here are details to account for the change of boundary conventions. At \(x=\kappa\), summing \(B_k\) over heights diverges, with starting triangle fixed, hence also the unrestricted susceptibility (each walk with its initial and final triangles specified incurs at most bounded multiplicity). At \(x<\kappa\), measure heights of triangle centers in units \(d_0/3\). A path starting from a fixed center and making a weak bridge (staying inclusively between the heights of its initial and final vertices, whose difference has absolute value \(k>0\)) has total mass at most \(C_x e^{-c_x k}\), using here \(x^{\#\text{steps}}\). Indeed for the increasing direction one can attach a bottom end on the bottom line of the initial vertex’s row, and a top end on the top line of the last vertex’s row. This uses no extra triangles if the triangle already has a ray in that boundary direction, and otherwise uses just an adjacent triangle strictly below or strictly above in its row, with a deterministic choice. The resulting walk is simple and a strip bridge, with known start and bounded multiplicity; thus the mass at \(\kappa\) was bounded, and there is exponential suppression by the minimal length. The decreasing case is reflected. A weak upward half-space walk from its lowest endpoint splits into such bridges by taking its last highest vertex, then in the remainder its last lowest vertex, and so on until exhausted. Their positive spans are decreasing strictly except possibly equality of the first two (consecutive heights along any single step differ). The total half-space mass is therefore finite by summability of the bridge bounds and \(\prod_{k\ge1}(1+C_xe^{-c_xk})^2<\infty\). Any unrestricted walk splits at a lowest vertex into two weak half-space walks rooted there; one may translate that vertex to fixed representatives of the two types, recovering the translation by the prescribed original start. This proves convergence. The two triangle types used here are equivalent by symmetry.

Flat and turned crossings

The following confinement estimate will keep the connectors in the cylinder construction and the later sewing argument inside prescribed channels.

A barrier estimate on the cylinder

Take the quotient by \(N\) horizontal lattice steps. We first construct essential polygons in slabs whose height is proportional to the period. Their positive mass will then control the vertical extent of other polygons and arches.

Passing to local nesting and bulk length

In the plane let \(Z(R)\) be the positive partition function (including empty) of mutually disjoint simple polygons surrounding a given triangular lattice point, within radius \(R\) of it, factor 2 per polygon, activity \(\kappa^{\rm length}\). We claim \[Z(R)=R^{1/12+o(1)}.\] Indeed on the even cylinder with antipodal marked points on the same level we computed \(\mathcal Z=N^{1/6+o(1)}\) for factor 2 on separating polygons. It bounds \(Z(c_1 N)^2\) above for sufficiently small \(c_1>0\) (that is, \(\mathcal Z\ge Z(c_1 N)^2\)). Conversely we can disregard essential polygons at bounded multiplicative cost in an upper bound for \(\mathcal Z\), since they must meet the joining segment and the sum of their single-loop weights is bounded. A contractible separator lifts uniquely around a specified preimage of the marked point on its inside, by simplicity (translated lifts have disjoint interiors). Hence its vertical extent straddles the common level. Those with vertical span above \(N^{1+\xi}\), for any fixed \(\xi>0\), are negligible in individual weight sum by the height tail. For the rest, if horizontal span in the lift exceeds \(N^{1+2\xi}\), project the specified lift around its marked point instead onto a cylinder tilted by \(60^\circ\) with integer period length between \(4 N^{1+\xi}\) and \(5 N^{1+\xi}\). It projects injectively even including the disk inside, by its narrow vertical span, and can be recovered from this polygon by the mark; it meets the level there of the mark and its span normal to the new period is of order at least \(N^{1+2\xi}\). The same height tail discards these as well. Disregarding all discarded polygons costs at most the exponential of twice the sum of their individual weights. All others split into the two types by the point inside, lifting to disjoint plane polygons within each type and supported up to radius \(C N^{1+2\xi}\); systems inject accordingly. Thus \(\mathcal Z\lesssim Z(C N^{1+2\xi})^2\). Both comparisons, monotonicity and then \(\xi\downarrow0\) give the claim.

The bounded-factor nesting estimate in the next section will turn this identity into the sharp first-length mass used in the exterior sewing argument.

Uniform geometric comparison

The height tails of the preceding section discard polygons much larger than the period, but their polynomial prefactor does not yet give a bounded cost at scale \(N\). We first bound the individual polygon mass in each planar diameter annulus. A refinement of the cylinder free-energy estimate will then leave only a uniformly bounded number of exceptional annuli.

Keep \(\beta=3/8\). Let \(D\) be a finite cylinder between flat caps, or a finite convex tiled polygon in the plane. Write \(Q_D=Z_H(\sqrt2)\) in the cylinder and \(Q_D=1\) in the plane, and use the same notation for restrictions of the available essential polygons. All these polynomials are positive. For a set \(K\) of occupied centers put \[H_D(K)=Q_{D\setminus K}/Q_D,\] where the numerator removes the essential polygons hitting \(K\). These ratios increase as the available polygon set grows. Indeed, for one deleted polygon \(v\) in a set \(P\), \[\frac{Q_P}{Q_{P\setminus v}} =1-t\kappa^{|v|}\frac{Q_{P\setminus N[v]}}{Q_{P\setminus v}},\] where \(N[v]\) is its closed conflict neighborhood. Induction on \(|P|\), deleting its neighbors successively in the smaller set, gives the claim; extra neighbors only increase the deletion ratio. Products give the assertion for several deletions. The same reasoning applies for \(0\le t\le\sqrt2\).

Write \(\mathcal Y(y)\) for the partition with these essential polygons and, in addition, mutually disjoint contractible polygons enclosing the triangular lattice vertex \(y\), each with factor \(2\). Expanding in the enclosing polygons leaves positive restricted essential partitions, so \(\mathcal Y(y)\ge Q_D\), including when \(y\) lies on a cap.

For a simple polygon \(L\subset D\) and a boundary source \(a\), let \(P(a,L)\) be the real part of the normalized signed mass of stems first hitting \(L\). Stop a stem at the incoming midpoint before visiting its first center of \(L\); include all essential polygons disjoint from the stem and \(L\), and divide by \(Q_{D\setminus L}\). The contour identity identifies this with \(1\) minus the real normalized mass exiting to the original boundary in the domain with \(L\) removed. This convention gives \(P(a,L)=1\) if the first center towards which \(a\) points has been removed. Since outer-exit real phases are positive and the deletion ratios increase before removal, \[ 1\ge P(a,L)\ge \sum_{\substack{\gamma:\,a\to\partial D\\\gamma\cap L\ne\varnothing}} \kappa^{|\gamma|}\cos(\beta W(\gamma))H_D(\gamma)\ge0. \tag{40}\]

Uniform analytic comparison

Return to the notation of Section 5; in particular \(I=\int_{\mathbb R}V(s)\rho(s)\,ds\) again denotes the bulk constant, not the complete-path current of the preceding subsection. We now sharpen the analytic estimate at the two values needed for the physical ratio. The precise output is convergence of normalized center values along finitely many residue classes. Saturation will make each of their limits positive.

We next transfer this bounded normalization to plane domains with explicit radius and lattice conventions.

Hexagon radii, offsets, and the chord estimate

First consider the Euclidean disk partition \(\mathcal W_y(t)\): every polygon surrounds the triangular lattice vertex \(y\) and is contained in the open disk \(B(y,t)\), with factor \(2\) per polygon and the usual critical activity. This partition includes the empty system. For a fixed sufficiently small \(c>0\), the uniform geometric comparison gives \[ \mathcal Z_{m,m}(1)\asymp\mathcal W_y(cN)^2, \qquad N=2m. \tag{51}\] Here the cylinder has horizontal period \(N\) and its marks are separated by \(N/2\). To spell out the comparison, two disk systems of radius \(cN\) about the two marks are disjoint and project injectively, giving the lower bound for the cylinder partition. For the upper bound, discard essential polygons and contractible polygons whose specified lifts have diameter at least \(cN\). The individual-weight bounds of Section 8.1 show that this costs at most a fixed factor: forgetting all avoidance involving a discarded polygon bounds the cost by \(\exp(2\sum_L\kappa^{|L|})\). Every remaining polygon lies in the disk of radius \(cN\) about the mark it encloses. Indeed a point inside a simple polygon belongs to its convex hull, so its distance from any point of the polygon is at most the polygon’s diameter. Separating the two marked types and dropping any avoidance between them gives the upper bound in (51). Changing \(c\) by a fixed factor changes only the constants, by the annular bound (44).

Combining (51) with the analytic estimate and interpolating monotonically between consecutive sufficiently large even periods gives \[ \mathcal W_y(t)\asymp(1+t)^{1/12}\qquad(t>0), \tag{52}\] uniformly in \(y\). For bounded \(t\) the lower bound follows from the empty system and the upper bound from a single fixed containing disk. Translation by a triangular lattice vector preserves all weights.

For clarity, the exact pure-sided regular geometry has no unspecified offset. In coordinates \[n_0=(0,1),\quad n_1=(\sqrt3/2,1/2),\quad n_2=(\sqrt3/2,-1/2),\quad n_{j+3}=-n_j,\] pure lines have levels \(d_0\mathbb Z\). If a regular hexagon has center \(v\) and inradius \(r\), its three support levels \(n_j\cdot v+r\) for \(j=0,1,2\) must belong to \(d_0\mathbb Z\). Since \(n_1=n_0+n_2\), the first and third levels minus the second give \(r\in d_0\mathbb Z\). Then \(n_0\cdot v,n_2\cdot v\in d_0\mathbb Z\), which says exactly that \(v\) is a vertex of the triangular lattice. Conversely every \(H(y;r)\) in the introduction has all six sides on pure cuts. Its inradius is \(r\) and its circumradius is \(2r/\sqrt3\).

The disk estimate gives a comparison that also covers nonregular roundings. If \(0<a\le b\) are fixed and \(B(y,aR)\subset D\subset B(y,bR)\), then positivity gives \[\mathcal W_y(aR)\le Z_D(y)\le\mathcal W_y(bR).\] Thus \(Z_D(y)\asymp_{a,b}R^{1/12}\) for \(R\ge1\). In particular it applies to every pure-support domain \[ D(y;\mathbf b)=\{z:n_j\cdot(z-y)\le b_j,\ 0\le j\le5\}, \quad b_j\in d_0\mathbb Z, \quad aR\le b_j\le bR, \tag{53}\] since it contains \(B(y,aR)\) and is contained in \(B(y,2bR)\). Some support inequalities can be redundant; the comparison does not require six active sides. Constants depend only on \(a,b\), not on the six support levels or the lattice translation. The marked point must remain a fixed fraction of the scale from the boundary for this two-sided bulk estimate.

Here is also the precise effect of translating and rounding a template. Suppose its center is \(v\) and its nominal support distances are \(r_j\). Replace each absolute level \(n_j\cdot v+r_j\) by any \(A_j\in d_0\mathbb Z\) with error at most \(K d_0\), where \(K\) is fixed. Relative to a marked lattice vertex \(y\), the resulting levels satisfy \[\bigl|b_j-\bigl(r_j+n_j\cdot(v-y)\bigr)\bigr|\le Kd_0.\] Whenever these levels satisfy the fixed inner and outer bounds in (53), the same comparison applies, uniformly over all the rounding choices. Thus a size parameter \(s\) can replace the actual radius \(r\) when \(a\le r/s\le b\) for fixed positive \(a,b\); equality of the two scales is not needed. The chord threshold below, however, uses the actual inradius.

The same quantitative argument explains the offset latitude needed later. If the six pure supports satisfy \(|b_j-R|\le R/20\), the horizontal cap height is between \(19R/10\) and \(21R/10\). The preceding tubes satisfy \[n_j\cdot(z-y)\le(21/40+\sqrt3/10)R<19R/20\] for the four tilted normals, so the lower chord mass remains uniform. Links with midpoints in \(B(y,R/4)\) are at distance at least \(7R/10\) from the boundary, while both adjacent lattice vertices have centered disks of radius \(R/2\) inside for large \(R\). The occupancy and first-length arguments are unchanged. These estimates cover the independent small relative support displacements and bounded row roundings in the sewing construction. They also hold with diameter threshold one half of the domain’s actual inradius, which is at most \((b_0+b_3)/2\le21R/20\). For bounded \(R\ge1\) there are only finitely many admissible support vectors after translating \(y\) to zero. The domain is convex and tiled, and both horizontal caps contain boundary ports. The same connected-dual-graph argument gives an opposite-cap chord of diameter at least \(b_0+b_3\ge19R/10\), so its mass and first-length mass are positive for either threshold. Taking the minimum over these finitely many domains extends the estimates to all such bounded scales as well.

The polynomial for marked edges

The aim of this subsection is to represent the positive mass of individual polygons through the two prescribed row edges as a contraction of the polynomial columns from Section 4. There is one extra fixed-spin slot at each mark in each column: in the notation below, the physical choice is \(k=2\), \(l=h=1\), and \(\delta=i\). At zero physical rapidities, the contraction divided by \(f_{m+n+2}^2\) will count each such polygon once. Here the interpolation parameter \(k\) counts the extra slots in each marked gap across the two columns; it is independent of the cylinder-period multiplier.

The fusion proof also needs the other distributions of these extra slots between the columns. Fusing an extra slot with a common slot removes both on that side and turns the partner common slot on the other side into an extra slot of opposite spin. Thus \(l\) or \(h\) changes by one, while \(k\) stays fixed. The closed family with \(k=2\) suffices for the two-edge application. We state the same identity for arbitrary \(k\) to give an exact contraction formula for larger fixed-spin lists; its proof uses the same recursion.

Write \(\Psi({\bf u})= f_{|{\bf u}|}({\bf u})w^i({\bf u})\) in its indicated site order, with \(\Psi(\varnothing)=1/\sqrt2\). Fix integers \(k\ge1\), \(0\le l,h\le k\) and \(m,n\ge0\). Take two orders ("below", "above") \[E_b, A, H_b, B;\qquad E_t, A, H_t, B\] where here \(A,B\) denote common lists of sizes \(m,n\), and the extra lists \(E_b,H_b,E_t,H_t\) have respective sizes \(l,h,k-l,k-h\). Each slot has a rapidity (physical variable), the same on common slots of course. Take the product of columns \(\Psi_b,\Psi_t\) evaluated on physical variables below and negatives of physical variables above. Contract common slots at equal spin \(p_j\), inserting \(\delta^{\sum_{j\in A} p_j}\) (\(\delta\ne0\)). Fix every spin on the four extra lists respectively to \(-,+,+,-\). The column lengths are \(m+n+l+h\) below and \(m+n+2k-l-h\) above. Denote the scalar thus obtained by \(Y\), with empty lists and products allowed.

We now give its finite sum explicitly. The common sites carry labels \(a\in\{-1,0,1\}\) and weights \(c_{\pm1}=\sqrt C\), \(c_0=2\). For an ordered common pair of rapidity difference \(u\) use the same-list or cross-list factor \(P_{aa'}(u)\) of Section 4, respectively; for a same-list pair also multiply by \[D(u)=\sin(u+2L)\sin(u-2L)\sin(u+b)\sin(u-b).\] Thus the pair and site factors have exactly the normalization of the unmarked polynomial. Sum labels \(a_j\) there subject to \(\sum_A a_j-\sum_B a_j=h-l\), and include the factor \((i/\delta)^{\sum_A a_j}\). Put \[J(u)=\cos(u+L)\cos(u+2L),\qquad (M^1_+,M^1_0,M^1_-)=(1,\cos(u+L),J(u)),\quad M^2_a(u)=M^1_{-a}(-u).\] For a common site of label \(a\) with value \(v\) and extra slot with value \(x\) include the factor \(M_a^{d}(x-v)\); on list \(A\) use \(d=1\) for positive extra spins, \(2\) for negative extra spins, and vice versa on \(B\). Outside the sum include \[Q_{lh}=\tfrac12 C^{k/2}s^{-k}(-s^{-1})^l s^{-h}\] and \(J(x_j-x_i)\) for each ordered-forward pair \(i<j\) in one of the extra lists below, or \(J(x_i-x_j)\) within either extra list above. Denote this product of within-extra-list factors by \(\mathcal J_{\rm ext}\).

For clarity, write the common site and pair product as \[\mathcal W(a)=\prod_j c_{a_j}\prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j) \prod_{\substack{j<j'\\j,j'\text{ in the same list}}}D(v_{j'}-v_j),\] where the common order is \(A,B\) and the kernel in \(P\) is selected by the two lists. Let \(d(j,e)\in\{1,2\}\) be the type just specified for common slot \(j\) and extra slot \(e\), and put \[\mathcal M(a)=\prod_{j\text{ common}}\prod_{e\text{ extra}} M^{d(j,e)}_{a_j}(x_e-v_j).\]

A direct estimate when the marks are antipodal

The homogeneous antipodal case gives a direct route from the physical pairing to the correlation exponent using the two-line gas already constructed. It displays the quotient by the zero separating-loop partition and derives the magnetic test potential, its scalar normalization, and the contour estimates reused below. We retain this calculation for those identities and for its independent homogeneous estimate, stated here at exponent-level precision. The later argument treats varying staircase rapidities and unequal intervals with uniform constants: when \(d\ll N\), two long intervals lie outside the short magnetic core, and their open operator product requires control that this homogeneous calculation does not provide.

On the homogeneous cylinder of even period with antipodal row sites, the polygon mass just identified satisfies \[K_2 \le (m+1)^{-4/3+o(1)}\qquad\text{(period }2m+2\text{)} .\]

Staircases and the two background lists

We now prove Theorem [u:two-edge-theorem]. Keep its notation \(P,d,k,N\) and the two marked edges. The insertion identity (54) is unchanged, but its two lists now have different lengths. The proof must keep the resulting two scales separate: a core of logarithmic length proportional to \(\log d\), and the whole period of length proportional to \(\log N\). All constants below are independent of \(P_1,P_2\) and \(k\).

Use a row cut obtained by a periodic monotone staircase in \(e_1,e_2\) (period \(P\), passing through the marks), and translate it in the third lattice-edge direction to make rows. This tiles by lozenges with acute or obtuse angle at the corner between the increasing row and the transverse vector; drawing as square tiles, the same honeycomb path weights are given by \(R(t+v_i)\) at argument \(L\) or \(2L\), respectively (the two choices reflect the base angle, interchanging \(a,h\) and \(c,d\) in the tile notation). By lattice reflection if necessary take the majority steps to have argument \(2L\). The infinite-height projection and orientation argument for the two marks in the flip formula therefore apply verbatim to these physical column parameters (above, their negatives in the formula by crossing). For the use of the projection, lifted individual arches have summable mass and through-lines to height tending to infinity have vanishing mass at fixed \(N\). Indeed in the planar lifted slab or half-plane bounded by these stairs the contour from a fixed boundary source has a common rotated real coefficient bounded positively: turns to other boundary exits are given by cyclic outward-normal differences minus \(\pi\), with a single consecutive pair of lattice normals for the rounding, so only one sign of excess up to \(\pi/3\) is possible. Transversal pure cuts can truncate the sides. For decay cut further by a pure hexagon of radius comparable to height. Mass to the hexagon cuts is \(O_N(r^{-1/4})\): average over order \(r/N\) translations by whole periods putting the starts at distinct positions in the central part of a common smaller hexagon (contour subtraction before translating), and reverse at the pure sides to use their escape bound in a positive contour, with order \(r\) such exits. This is the flat-side argument of the geometric estimates. Thus through-line systems disappear by absolute single-path bounds and empty projections have the same limits.

For detail use lattice coordinates \((X,Y)\) along \(e_1,e_2\), with transverse step \(e_2-e_1\). Between transverse truncations (bounds on \(X+Y\)) and the two periodic stairs this domain has lower and upper bounds on \(Y\) as functions of \(X+Y\), both with slopes 0 or 1. Intersecting with pure hexagons or parallel pure half-planes (bounds on \(X,Y,X+Y\) at lattice levels) retains that same form: the bounds use maxima/minima, with feasible section on an interval since comparison with each affine bound uses monotonicity (of both stair coordinates). The CCW tangent directions thus occur in a lower block alternating directions 0,1, then possibly direction 2, upper block using 3,4, possibly direction 5 (units \(\pi/3\)). The cyclic tangent increase to exits from a given boundary midpoint therefore stays in either \([-\pi/3,2\pi]\) or \([0,2\pi+\pi/3]\), depending on the starting direction. Subtract \(\pi\) to get the path turn by boundary closure. This gives the asserted rotated positivity since \((3/8)(2\pi+\pi/3)<\pi\), and applies to subtraction of signed contours as before (terms on shared exits which stay inside agree). The planar flux rule still applies in these simply connected regions. To bound through mass from one stair in a wide truncation at fixed period, take a common lattice hexagon centered near that start of radius comparable to slab height but small enough not to reach the opposite stair. Start also at its period translates in the central part, and subtract the contour in the intersection to bound the individual through masses by the mass to the hexagon cuts. Upon reversal from each such new exit, cut again parallel to its pure side before the central targets; subtraction bounds their mass by that to the new line, at most a constant times the straight strip bridge sum at comparable scale. Letting the transverse truncations recede, the through masses are equal by periodicity. This gives the bound stated above. Arch sums themselves are bounded just by the positive contour, using exhaustion.

Both extra rapidities in each column now have the same value, denoted \(x_e\). The two common lists have sizes \(m=d-1,n=N-d-1\). Augment each for the plasma calculation by one site at \(x_e\), forming background lists of sizes \(M_A=d,M_B=N-d\), both even, now having identical proportions of the two possible values. In this section use \(\lambda_b=\pi/(2b)\); write \(\lambda\) for the proportion of minority steps, \(0<\lambda\le1/2\). By translation of rapidities we take the majority value to be \(r_0 L\) and the minority \((r_0-1)L\), where \(r_0=1/4+\lambda/2\).

The residue calculation starts from all common labels positive (whether neutral then or not), using its product as prefactor. Exactly as in the balanced two-leg case the potentials are now one copy of \(R_{g h}(i(w+i v_e))\) per augmented-list site \(e\) of color \(h\), value \(v_e\), and the common magnetic factor \(-\sinh(w+i x_e-iL)/\sinh(w+i x_e+iL)\); this follows by translation in each extra-slot factor of that calculation. Denote the remaining residue sum by \(P'\). The prefactor divided by \(f_N^2\) in the observable is \[ \frac{C^{N/2}\prod_{e<f:\ g_e=g_f}D(v_e-v_f)}{2 f_N^2} \prod_{e\text{ common}} \frac{J(v_e-x_e)}{J(x_e-v_e)} \tag{55}\] by \(D(y)=J(y)J(-y)\), \(Q_{11}=C/2\). Elementary lowering counts satisfy \(K_g=M_g+u\) with common integer \(u\).

Periodize as before, using magnetic factor \[U_T(w)=-\frac{S_T(w+i x_e-iL)}{S_T(w+i x_e+iL)} \left(e^{-3iL}\frac{S_T(w-T/2+iL)}{S_T(w-T/2+4iL)}\right)^2,\] and in the central theta sum include background charges \(q_e=-q_{g_e}\) at offsets \(a_e=-v_e\) (to avoid confusion, \(q_{g_e}\) itself denotes the color sign as for a particle). Thus \(X\) in the center Gaussian includes \(\sum_e q_e i a_e\). This ensures periodic differentials. Initially use rapidities near zero, so all contour openings and deformations work by continuation. The lift is still a finite sum analytic near nome zero, by split poles as before. At final rapidities differences \(L\) in the site lists one uses limits: before opening, the individual residue terms obtained by splitting have only denominators from the displayed shifted \(S_T\)’s and constants, independent of counts of sites except via multiplicities. Possible singularities at exact differences stay on the very same pair hyperplanes independently of nome on the small disk. Removability there for the lifted sum will also follow below by analytic continuation in rapidities of the final integral bounds (identity first for real periods). Thus maximum modulus at those values is permitted.

The ordered magnetic gas

The staircase determines the background lists. We next express their residue sum as an ordered gas whose integer height records the winding. Keeping the phases in this step is essential: the interval estimate below uses the resulting transfer operator, not the absolute value of an unidentified periodic partition function.

Here are details of extending the mode calculation. Put \(E(p)=\sum_e e^{-p a_e}{\bf e}_{g_e}\). In the mode matrix include extra background slots \(E_e\) with count-mode 1, pair entries from the potential factors (so \({\cal M}_{R,E}n_E=-K\widehat\rho E\)), inverse couplings to them initially zero. Their mutual block is zero except that formally at Laurent zero assign the pole terms \(4q_eq_f/p^2+4q_e q_f(a_e-a_f)/p\), of zero finite part. Then:

• The bulk term in log is \[ S(T)=\frac1{2T}\sum_p E(-p)^t K(p) E(p)\widehat\rho(p)^2 -\frac12(T/3-2\log2)(M_A-M_B)^2. \tag{56}\] Determinants and measure conventions in \(\eqref{eq:legacy-P-star}\) are unchanged.

• Chemical potentials per final particle (before adding the magnetic test) are \[ 2\pi i\,\sigma_i\sum_{e:g_e=g_i} \int_0^{s+i(v_e+\delta_i L)}\sum_{\nu\in\mathbb Z}\rho(z+\nu T)\,dz , \qquad \sigma=(0,1,1,-1,0). \tag{57}\] Fugacity \(e^{-u r}\) if used still brings \(T r_j^2/(4h_*)-r_j(u_*+\bar L)\). Indeed distribute \(2X^2\) into all pairs including background-background. The terms with \((a_i-a_j)^2\) in units of \(1/T\) again sum to \(-2 A_{\rm tot}^2\) with all charged offsets included; the missing constant logs from the background block give just the second term of (56). Thus the same verification by \(\gamma\) and its two charge conditions works with the extended block, including the center terms with \(l\); the Laurent saddle adds \(\widehat\rho E\) to the earlier residual saddle, giving the same common-integer Poisson sum and (56). In the chemical calculation the old simplifications apply in both color sectors individually: with \(H_V\) defined like \(H\) using multipliers \((0,-1,0,0,2c,0,0,-1)\), one has \((H_V+H\widehat\rho)(a;t)\) at \((0;0),(2;1),(2;-1),(3;-1)\) respectively \(0,0,\widehat\rho,0\). Insert the rapidity shift exponent to obtain (57). These formulas are for analytic continuation, so \(\sigma=0\) terms stay absent even beyond a primitive strip.

More explicitly at zero take the raw right count vector with left and inverse modes as before, appending background modes identically 1, and prescribe the same derivative condition \(q^t\gamma'_{\rm raw}=A_L+\sum_e q_e a_e+\pi l/2\). Then the second-order and first-order poles of the extended matrix still have the displayed charge/offset/winding form of the gas proof, so evaluation gives the actual center Gaussian and winding phases (including phase slopes from the potentials on left contours). The sum of charge products for pairs including real charges but not background-background alone is \(-\tfrac12(M_A-M_B)^2\) minus half the elementary count; regular diagonals treat the latter part as before. The additional saddle term \(\widehat\rho E\) neutralizes the appended charges at 0 and contributes exactly \(\sum_e q_e a_e\) to the derivative charge sum, while its count value \((M_A,M_B)\) leaves the free integer and Gaussian shift unchanged. The assigned background block has zero contracted finite part of its own. For (57) the remaining linear symbol at a particle against each same-color site is \(-2\pi\sigma_i e^{(v_e+\delta_i L)p}\widehat\rho(p)/p\), yielding that primitive including the finite part at zero.

We describe the ordered-position form of the residual sum. Let \(\ell_{gi}\) count original left elementary members of species-color index \(i\) belonging to \(g\), write \(\ell_{\Sigma i}=\sum_g\ell_{gi}\). For particles in increasing order of real contours, write \(H(s)=j+\sum_{\Re s_d>\Re s} W_d\), \(j\) the Poisson integer; at a jump use its mid-value \(H_m\). Winding neutrality closes this integer path. Set the real row \(c^t\) by \[{\bf1}^t(J(p)-K(p)\ell)=2\pi W^t/p-2h_* c^t+O(p).\] (In this expression \(J\) is the coupling block of the transform, transported on the final constituents, not the site function in (55).) Thus \(u_*+\bar L=c^t n-(\pi i/h_*)\sum W_d s_d\). At zero, \[ Z=\alpha WW^t/p^2+M_*/p+O(1),\qquad M_*=-2\pi(\widetilde{\cal W}+W c^t-c W^t), \tag{58}\] where \(\widetilde{\cal W}\) is an integral skew matrix. Indeed change right variables by \(-\ell n\) in the block matrix before Schur. The non-right charges are now zero, its non-right block has simple coefficient \(-2\pi\widetilde{\cal W}\) from the shifted winding matrix (\({\cal W}_{nn}-\ell^t{\cal W}_{Rn}-{\cal W}_{nR}\ell\) before subsequent branch corrections). The remaining poles after Schur come just from the common channel as shown. Exact-offset averages for free left clusters give integral windings here (symmetric shifts for same offset clusters; otherwise all off singular offsets, even sum of multiplicities).

Apart from the explicit terms of \(\eqref{eq:legacy-P-star}\) and site/test couplings the constant phases, when combined with the ordered factor from \(M_*/p\) and the term \(-2\pi i j c^t n\), reduce up to a single unit prefactor to \[ \prod_d \eta_{i_d} \exp[-2\pi i c_{i_d} H_m(s_d)] \tag{59}\] for fixed unit constants per species, the same for every choice of even \(M_A,M_B\), including 0. Here and below indices in such products include the color. To check this, the \(1/p\) kernel at ordered separation is \(-i/2\) (positive separation), so neutrality gives the stated mid-value term and an unordered sign \((-1)^{\widetilde{\cal W}_{ab}}\) per pair. Before branch moves, original log constants have signs \((-1)^{{\cal W}_{ab}}\) on pairs of left particles, plus \((-1)^{\sum k_g P_g}\) from right-left pairs, where \(k_g=M_g+u-\sum_i\ell_{gi} n_i\) and \(P_g=({\cal W}_{R,L'} n_{L'})_g\); backgrounds contribute no sign (zero right winding, integral left windings and even counts near zero rapidities). Indeed for each shifted product of sinhs the positive-separation leading logs have just the offsets times \(i\), without extra signs before taking the indicated means and jump, since the products use symmetric shifts. Residue orientations can add single-cluster phases only. Original right and inverse same-side symmetrization gives exponents \(\sum_g\binom{k_g}2+\sum_{g,t}\binom{n_{gt}}2\), and the Weyl exponent adds \(\sum_g\lfloor(1-k_g-n_{I,g})/2\rfloor\), with \(n_{gt}\) inverse counts, \(n_{I,g}=\sum_t n_{gt}\). These are exactly the constant-mode signs in the trace calculation of \(\eqref{eq:legacy-P-star}\) (between different ordered oscillator radii the geometric log has zero constant phase). Dependence on \(u\) cancels since each simultaneous unit increment flips \(\sum_g(P_g+n_{I,g})=0\bmod2\). At \(u=0\) changing \(M_g\) by 2 also has no effect. Now on count space all second differences of this total sign exponent, mod 2, are precisely \(\widetilde{\cal W}\): in addition to the left-left and right-left terms, the binomials and floors (using \(\lfloor(1-h)/2\rfloor=\binom h2\bmod 2\) for integral \(h\)) contribute per group the polarization entries of shift type \(\ell_g\otimes n_{I,g}+n_{I,g}\otimes\ell_g+n_{g+}\otimes n_{g-}+n_{g-}\otimes n_{g+}\) (symbols here denote linear count functions). These give the inverse-inverse and shifted right-inverse windings. This cancels the unordered signs from (58), leaving just single-particle phases. Each subsequent branch change of order \(\nu\) shifts \(\widetilde{\cal W}\) correspondingly and brings the matching constant sign \((-1)^\nu\) between the pair by the ratio of geometric factors as in the mode calculation. Fusion restricts to sums of constituents (self-residue has only a local extra phase), with the same compensation for exterior pair branch changes. This proves (59).

For the short part of pair energy we use the periodization of \(G_{ab}(s)\) for ordered positive separations (target to the right, source to the left), with the usual regular diagonal at contact and both periodic continuations counted, once per positively separated pair up to translations. This denotes the inverse Fourier kernel after removing exactly the two pole kernels in (58), with transpose at negative separations. Thus the off-contact exponent in the gas is minus one copy of \(G\) per such pair, including pairs with translated particles, and the regular contact self term with half weight. For \(\Re s>0\) we have \[ G_{ab}(s)=\sum_{l\ge1}\sum_{k=0}^{r_*} C_{ab,l,k}s^k e^{-\omega_l s}, \tag{60}\] where \(\omega_l>0\) increase at least linearly, \(C\) grow at most polynomially and \(r_*\) is fixed. Indeed the rational functions in the final symbols of the two-line calculation have only unit-circle poles in \(e^{Lp}\), of uniformly bounded order (the extra denominators after Schur come from \(H_*\), proportional in the channel symbol to \((1-e)(\coth(bp)-c\,{\rm csch}(bp))\), and use the earlier factorization of \(1-e\)). Shifting the inverse contour downwards in imaginary-period multiples picks these residues apart from the already separated zero pole terms. The horizontal ends can be bent down at small slope (symbols subtracted at zero are bounded or \(O(1/p)\) there); equivalently for the unsubtracted meromorphic symbol use a principal finite-part integral, whose zero pole contributes just the two displayed long kernels. Along deep level shifts chosen a fixed distance from poles the tail rotation yields exponentially vanishing remainder. This agrees at real periods with the ordinary periodized short kernel by Fourier inversion (at contact the one-sided short parts can have a jump; for the self entry there is no jump), and then by the continuation of \(\eqref{eq:legacy-P-star}\). More explicitly removing \(\alpha WW^t/p^2+M_*/p\) gives a symbol regular at zero, \(O(1/|p|)\) at either end, with inverse integrable at zero (log plus bounded terms). Its ordinary periodization includes the exact zero harmonic; the pole kernels separately are as in (28).

From periodic gas estimates to interval operators

We next construct an operator whose products describe intervals of the ordered gas. This construction is needed because the magnetic core and the two exterior intervals have different particle potentials. A bound on homogeneous closed periods would not by itself control their product.

Here is the required transfer estimate. Fix a large cell length \(S_0\); consider first a homogeneous gas with no magnetic functions or (57), but take \(r=-2a_0'\) constant, \(a_0'=+6Li\) or \(-6Li\), and allow all five species of color \(A\), only \(S,F\) of color \(B\). In addition to \(G\) and (59) this has exponent \[ -h_*^{-1}\int(\pi H+i a_0')^2\,ds + 2a_0' c^t n. \tag{61}\] The construction below will give a common state space for the two tilts; Lemma 16 then states the strict spectral bound.

Localization of the magnetic source

The homogeneous interval operator is now controlled. We turn to the actual magnetic source, prove the uniform one-particle estimates needed to compare its tails with that operator, and retain the suppression inside the shorter core. Write \(a(s)=\log U_T(s)\) on the right contours, using the branch whose bulk core log comes from the odd primitive in the centered argument \(s+i x_e\); its total winding around the period is zero as in the balanced calculation of Section 9.2. All logarithms in exponents are continued along the stated graphs. As there, the right mean of \(a\) gives \(r=-2\bar a\) and scalar \((M_A+M_B)\bar a\); on the left retain direct \(U_T\) factors and subtract the mean, on the right transform insert \(a-\bar a\). Completing Cartan nonzero modes exactly as in the balanced case gives the following additional external scalar (in addition to (56), with the squared linear-primitive effect assigned below): \[ S_a(T)=\frac1T\sum_p \widehat\rho(p){\bf1}^t E(-p)\widehat a(p) +\frac1T\sum_{p\ne0}\big(Q_+(p)-1/h_*\big)\widehat a(-p)\widehat a(p). \tag{63}\] The local height-square exponent is now again \[ -h_*^{-1}\int (\pi H+i a(s))^2\,ds . \tag{64}\] Indeed the added particle-source nonzero-mode multiplier is \[-J(-p)^t{\bf1}Q_+(p);\] its simple pole \(+2\pi W/(p h_*)\) acting on \(a-\bar a\) provides the corresponding mean-zero primitive of \(-i(a-\bar a)\). Summing jumps evaluates it by parts against \(H\); remaining terms here are exactly the long quadratic from (58), mean/fugacity tilt and the part of the test variance not kept in (63). The extra exponent at each particle, besides (59), consists of: \[ (2c_i-\ell_{\Sigma i}) a(s)+\sum_{\text{orig. left members there}} \log U_T(s+i(\delta_i+b_{i'})L)+ b_i'(s). \tag{65}\] Here offsets \(b_{i'}\) denote the indicated individual physical integer additions. The short source \(b_i'\) is the Fourier action on \(a'\) of \[[-J(-p)^t\allowbreak{\bf1}Q_+(p)-2\pi W/(p h_*)-(2c-\ell_\Sigma)]_i/(-ip),\] regular at 0. Indeed the definition just before (58) gives the indicated constant coefficient \(2c-\ell_\Sigma\) using \(Q_+=1/h_*+O(p^2)\). Its mean times the counts comes from \(-\bar a \ell_\Sigma^t n+2\bar a c^t n\) in the left and fugacity terms; likewise the tilt with the mean of \(a\) in (64) uses \(\int(H-j)=\sum W_d s_d\). Direct left log terms just mean products with their integer multiplicities.

Keep the cell length \(S_0\), block length \(q\), and spectral margin chosen above fixed. Choose the bend size \(\theta_0\) sufficiently small, then the transition length \(A\) and buffer \(g\gg S_0\) sufficiently large to meet the slope bounds below. The fixed boundary layers used in the product estimate may depend on these choices, but none of these constants depends on \(d,N\), or \(\lambda\).

Take nominal period \(R=2a_0^{-1}\log N+g+O(S_0)\), a multiple of \(2S_0\), with \(a_0=\pi/b\) as in the gas calculation. Allow \(|T-R|<\pi+1\). Use a common curve for the positions parameterized by \(s=t+i y(t)+(T-R)\chi(t)\), \(-R/2\le t\le R/2\), with the analytic period adjustment \(\chi\) as in (28), derivative \(O(1/g)\), supported near the ends within \(g/3\), \(\chi(\pm R/2)=\pm1/2\). Use the periodic, real, Lipschitz-small displacement \(y(t)\) zero for \(|t|\le A\), decreasing in \(|t|\) to a fixed tiny negative constant \(-\theta_0 L\) by \(|t|=2A\), then flat (\(A\) large absolute, \(\theta_0>0\) sufficiently small). Slopes meet all earlier smallness requirements.

Here are uniform bounds on site sources on this graph:

1. \(a(s)\) tends exponentially to \(a_\pm=\pm6Li\) away from 0 and the period edges, on \(\pm t>0\), and is bounded. Logs in (65) from direct multipliers likewise tend to these phases modulo \(2\pi i\), with multipliers bounded. The short \(b_i'\) tends exponentially to 0 there with \(\exp(\Re b_i')\) bounded throughout.

2. For \(D,I'_+\) the real exponent in (57) is bounded above by \(-c_0 M_g e^{-a_0|t|}+O(1)\) with \(c_0>0\); the same estimate holds for \(I'_-\) when \(|t|\ge 2A\), and an upper bound \(O(1)\) everywhere. Moreover for color \(A\) the (57) multipliers tend uniformly to 1 far outside \(|t|\le a_0^{-1}\log d\) and away from the period edges.

For 1, the product formula and derivative periodizations about the two centers are as in the balanced case. Direct left physical offsets have no poles along the graph (the edge compensator unshifted in rapidity). For the short term apply the regular multiplier on the two separate whole-line coth derivative differences before periodization; their transforms off zero decay at the nearest respective shifted singularity, by integrating across one coth period. Products for the core have exponential decay in frequency (also on slightly bent shifted contours for spatial transport) uniformly in the shifted rapidities: the rates are controlled by \(1\mp(x_e/L+\delta_i)\); in a limiting case at \(S\) with positive frequency use the additional exponential decay of \(A_0(e^{Lp})\) there. For \(I'_-\) with negative frequency, isolate from \(b_i'\) for this core part only the multiplier \(-(e^{\delta_i Lp}-1)/(-ip)\). This gives minus the core log difference on shifting the argument by \(i\delta_i L\); its exponential modulus is bounded above (a possible approach is to the pole, not the zero, of the shifted ratio) and it is localized exponentially including its images. After this subtraction the extra exponent has uniform frequency decay since \(A_3=-1\). These statements use \(a'\) itself split as periodized whole-line differences around the centers (the opposite masses cancel globally). Contour rotation away from the centers as in (28) gives all asserted periodized bounds and continuation; for the isolated log difference use direct transport, with the limiting approach to a singularity only near \(t=0\) where the graph is undisplaced. Widths for the unshifted ratios in such estimates are bounded away from singularities since \(-3/4\le x_e/L\le1/2\).

For 2, images and period adjustments cost \(O(1)\) by exponential decay and the choice of period. On the basic line the imaginary part for a single primitive at argument \(t+iLY\) is \[B(u,Y)=\frac1{2\pi}\operatorname{atanh} \frac{4\sqrt3\cosh u\sin(\pi Y/3)}{3+4\sinh^2u+4\sin^2(\pi Y/3)},\qquad u=a_0 t ,\] by integration of the explicit density. Before bending, take the mixture of \(Y=r_0+\delta_i,Y-1\) with proportions \(1-\lambda,\lambda\). For \(I'_+\) its sign is uniformly positive with size at least constant times \(e^{-|u|}\): \(Y\ge2/3\) and the positive function increases on \(0<Y<1\). For \(D\), \(Y=1/3+\lambda/2\); the same follows by monotonicity and continuity (including as \(|u|\to\infty\) after scaling) if \(\lambda\ge1/3\). Here are convenient comparisons for the remaining cases: on \(0<Y<1\), \(2\pi B\) divided by \(4\sqrt3\cosh u\sin(\pi Y/3)/(3+4\sinh^2 u)\) lies between 1 and \(\operatorname{atanh}(\sqrt z)/\sqrt z\), \(z=4\sin^2(\pi Y/3)/(3+4\sinh^2u)\). Indeed compare the argument of atanh to \(\sqrt{3z+z^2}/(1+z)\) below (the atanh at this lower value is at least \(\sqrt{3z+z^2}\) by differentiation, and atanh divided by argument increases) and \(2\sqrt z/(1+z)\) above. Thus for \(D,\lambda\le1/3\) the required margin follows from \[(1-\lambda)\sin(\pi Y/3)-1.33\lambda\sin(\pi(1-Y)/3)>0\] The split at \(\lambda=1/4\) makes this a quantitative comparison. For \(0\le\lambda\le1/4\), use \(\sin(\pi/9)>1/3\) and \(\sin(2\pi/9)<2/3\); the displayed difference is at least \(17/600\). For \(1/4\le\lambda\le1/3\), use \(\sin(11\pi/72)>9/20\) and \(\sin(13\pi/72)<11/20\); it is at least \(337/6000\). The factor \(1.33=133/100\) is valid throughout these comparisons: the relevant \(z\) is at most \(5/9\), and \[\frac{\operatorname{atanh}\sqrt z}{\sqrt z} \le 1+\frac z3+\frac{z^2}5+\frac{z^3}{7(1-z)}<\frac{133}{100}.\] For \(I'_-\), \(Y=\lambda/2\le1/4\), so \(z<1/10\) and the same series bounds the upper-comparison factor by \(11/10\). Moreover \[\frac{11}{10}(1-\lambda)\sin(\pi\lambda/6) \le \frac{11\pi}{60}\lambda <\lambda\sin(\pi/4) \le \lambda\sin(\pi(1-Y)/3).\] This yields the negative sign with a fixed margin times \(\lambda e^{-|u|}\). At large \(|t|\) bending down only improves that comparison, giving an absolute margin at full bend: \(B/e^{-|u|}\) there is uniformly increasing with slope bounded below on the needed interval by the displayed formula. Other comparisons are unaffected for small \(\theta_0\). This proves the suppression bounds. The real-axis primitive limits to \(\pm1/2\), unchanged on bounded parallel displacements with exponentially small error; in the last assertion of 2 the masses \(M_g\) are even. These arguments use analytic continuation on horizontal graphs first (\(|r_0+\delta_i|,|r_0-1+\delta_i|<1\) for nonzero \(\sigma_i\)) and the bend only far from core poles.

These estimates justify the mode transform at the indicated rapidities. More explicitly start at real large \(T\), near zero rapidities, where the analytic test can be taken through the original trace and the final fusions without pole problems, exactly as in the balanced case. Continue rapidities linearly to the targets on the final contours with horizontal projections (no remaining branch crossings in the continued (57) or magnetic terms); absolute convergence throughout and in neighborhoods for fixed parameters follows by (28) or (62), since only bounded one-body and at most linear height perturbations are added. For the lift, split poles initially near the fictitious background only for actual remaining residue sites, as in the balanced magnetic calculation (including common sites possibly colliding with those extra values); this gives meromorphic continuation with a finite product-sum representation. The straight continuation just given implies regularity there on the rapidity family used, even at collision divisors where the individual split terms need limits. Now deform the final position graphs to include the small \(y\) at real periods by simultaneous transport (contact repulsions have fixed continuation on the shared small-slope curve; long ordered factors can equivalently be reassembled in their analytic pair branches; no one-body singularities are crossed). This can also be done termwise before resumming the Poisson index by using each original Fourier coefficient for periodicity. Equivalently the Coulomb and mid-jump rules are local on the cyclic integer path, so after summation boundary terms always reindex periodically. Coincident loci have no poles, only possible zeros of nonnegative integral order with analytic transport, by the high-frequency coefficients. Uniform absolute convergence on the deformation follows as in (28). Holomorphic continuation in the period disk then holds on the indicated parametrization, locally also for rapidities near the targets. Any singularities from split residues in taking those target rapidities at finite nome are on fixed rapidity divisors by the product rules at large \(\Re T\), hence the real-period regularity removes them identically near nome zero (analyticity in nome of the desingularized finite expression). Its value at zero is the original trigonometric sum, likewise by continuation. We thus bound on the nome circle using this common graph. Parameters can be kept on a full fugacity period when coefficient verification is needed.

A bounded product through the core and tails

Choose \(u_0\) a nonnegative cell-boundary coordinate within bounded distance of \(a_0^{-1}\log d\). On the core \([-u_0,u_0]\), where \(\chi=0\), remove the scalar \(h_*^{-1}\int a^2\) from (64), of real part \(-2(2u_0)+O(1)\), since \((6L)^2/h_*=2\). Redistribute the principal real linear tilt on that interval using \[\int_{-u_0}^{u_0}\operatorname{sgn}(t) H(s(t))\,dt =-\sum_{d':\, |t_{d'}|<u_0} W_{d'}(u_0-|t_{d'}|).\] It appears with positive coefficient \(12\pi L/h_*\); terms from purely imaginary tangents here have unit modulus for this leading tilt. The unfavorable sites all have \(W<0\) and thus the strong chemical suppression, which absorbs their loss at bounded additive cost per particle while retaining an overall suppression \(-c_1(u_0-|t|)\) per particle there (\(M_g\ge d\), and no species has \(W=0\)). Here we actually replace that real part of the tilt in the local edge weights on the core by the point terms from the displayed identity; their cyclic product is unchanged. All other added terms have bounded-above point log moduli by 1 and 2, and at most bounded linear height costs on cells, whose real coefficients beyond the reference Gaussian tend to zero away from the center after this redistribution. There is no winding boundary cost in this telescoping.

Every edge kernel is uniformly bounded after the scalar removal. We need the stronger assertion that the product through the whole core has bounded norm, independently of its length. We prove it directly. For a core cell let its depth be its minimum distance from the two core cuts, truncated below at zero. Reserve a fixed small fraction of the remaining negative point potential \(-c_1(u_0-|t|)\): on cell \(i\) this gives an outgoing factor \(\exp[-p_i(x)]\), where \(p_i(x)=\delta\,\mathrm{depth}_i N_x\) for a fixed \(\delta>0\). Set \(p_i=0\) on the exterior states at the two core interfaces. If \(K_i(x,y)\) is the outgoing edge kernel, replace it by \[K_i^{\rm sym}(x,y) =\exp\!\left[\frac{p_i(x)-p_{i+1}(y)}2\right]K_i(x,y).\] The reserved factor on this edge becomes \(\exp[-p_i(x)/2-p_{i+1}(y)/2]\). In a product through the core the multipliers telescope pointwise, including the interfaces where the exterior factors are one. Thus the complete product kernel is unchanged; no boundedness of an inverse diagonal gauge is required. Adjacent cell depths differ by at most the fixed cell length. The unused point suppression and the Gaussian margins in (62) therefore remain available on both incident edges. In particular, every block with a nonempty state at either endpoint has operator norm at most \[C\exp(-c\,\mathrm{depth}),\] where \(\mathrm{depth}\) is the smaller of the two incident cell depths. Here an empty state means no particles in the cell; its integer height and Fock state are still retained.

Away from a fixed number of cells at the central transition and the two cuts, the empty-to-empty block has norm at most one. Indeed the integer height is constant across such an edge and the real reference exponent is \(-\pi^2 S_0 H^2/h_*\). The residual linear coefficient of \(H\) tends exponentially to zero away from the transition and the cuts. Since \(H\) is an integer, its absolute contribution is then bounded by the negative quadratic term whenever \(H\ne0\); for \(H=0\) both contributions vanish. The common tangent is flat there, the remaining scalar phases have modulus one, the state gauge cancels, and the free Fock propagation is contractive. The complete edge consequently has norm at most \(1+C\exp(-c\,\mathrm{depth})\) on each of the two core halves outside those fixed exceptional layers. The sum of these errors is bounded: for each depth there are only a bounded number of cells. Taking the product proves a uniform bound on the core operator. The exceptional layers contribute only fixed factors. We use the redistributed gauge only on core cells; the tail operators retain their original gauges.

On each of the two tails outside the core, far from its end cuts and the period edges, (64),(65) and (57) converge in the same edge-kernel sense, uniformly, to the homogeneous kernels with (61) and phases (59), since all undesired \(B\)-species are suppressed, and points on \(A\) see the chemical limit 1. The direct left factors with the displayed linear term in (65) give exactly \(\exp(2a_\pm c_i)\). Pair kernels are unchanged on the interior flat cells (their common displacement cancels). Convergence in Hilbert-Schmidt norm as distances increase follows by bounded-state truncation and the Gaussian majorants. Use the fixed block length \(q\) chosen after Lemma 16, for which each homogeneous \(q\)-fold product has norm at most \(\exp[(1/2-\varepsilon)qS_0]\). Increase the fixed boundary layers until the \(q\)-fold products of the actual tail edges differ in norm from their homogeneous products by less than the available fixed margin. Their norms are then at most \(\exp(qS_0/2)\). The boundary layers and the fewer than \(q\) remaining edges have bounded length and bounded norms. Consequently the two tails together contribute at most \((R-2u_0)/2+O(1)\) to the logarithm of the product norm. Short intervals and all remaining boundary layers need only the uniform edge bounds. In the cyclic trace estimate keep two edges in Hilbert-Schmidt norm to use the remaining product bounds. Including the core scalar but excluding (56),(63) and determinant, we obtain overall log modulus at most \[ -2R+\tfrac52(R-2u_0)+O(1). \tag{66}\]

Matching the scalar and the Pfaffian normalization

The interval bound controls the particle integral. To recover the polygon observable we must still match its scalar factors with the normalizing polynomial \(f_N^2\). This step is necessary even when the staircase uses only two rapidity values, because those values may have arbitrarily unequal multiplicities.

We first match the external scalars. Uniformly on the circle, (56) equals \[\tfrac12\sum_{e,f} I_{g_e g_f}(v_e-v_f)+C_2 e^{-2T}+O(1)\] for a constant \(C_2\) depending on the site data. Here \[I_{gh}(v)=2 q_g q_h\log2+\int_{\mathbb R}\! [K_{gh}(p)\widehat\rho(p)^2\cosh(vp)-4q_g q_h/p^2]\,\frac{dp}{2\pi}.\] Indeed the subtracted pole all-mode sum is \(q_g q_h T/3\). The ordinary regular-symbol periodization picks corrections from poles at \(\pm2i\) and an error \(O(N^2 e^{-a_0 \Re T})\), by contour shifting with exponentially bounded tails; there is at most a simple pole at \(\pm i a_0\) also in the color difference channel by substitution. We have \[ e^{I_{gh}(v)}= B(v)^2\tan^2(\lambda_b v) \begin{cases}1/D(v),&g=h,\\ 1,&g\ne h,\end{cases} \qquad B(v)=\sin(v+b)\sin(v-b)/\sin v . \tag{67}\] To verify, first take the difference from \(v=0\). Indeed with \(t=e^{-Lp}\) the multiplier \((p/2\pi)K_{gh}\widehat\rho^2\) on \(p>0\) is \(t^2/[(1-t+t^2)(1-t^8)]\) times \(1+t^6\) or \(-2t^3\) on same or opposite color respectively. Thus it expands as \(\sum_{k\ge1} c_k e^{-kLp}\) with \[c_k=-2{\bf1}_{k=\pm3(8)}+2{\bf1}_{k=0(8)}+2{\bf1}_{k=3(6)}-2{\bf1}_{k=0(6)} +{\bf1}_{g=h}{\bf1}_{k=\pm2,\pm3(8)} .\] Integrating gives the sum of \(-c_k\log(1-v^2/(kL)^2)\), exactly the log ratio by the sine product. To fix the constant use \(v=i V\), \(V\to+\infty\) along the analytic convolution strip: by the pole transform and the integrable regular remainder \(I=2q_gq_h(\log2-V)+o(1)\), matching (67).

Formula (63) equals \[\sum_e\log\frac{J(x_e-v_e)}{J(v_e-x_e)} + C_1 e^{-T}+O(1).\] The regular test variance part costs \(O(1)\) as in the balanced magnetic calculation of Section 9.2 (\(a'\) localized and analytic near the right graphs). The cross part integrates \(a\) against the shifted periodized densities (here one can use the centered straight period segment, slightly sloping for complex periods). The log of the normalized squared edge ratio gives first terms \(e^{-T}(d_1 e^{2s}+d_2 e^{-2s})\) across the core with constants \(d_1,d_2\), with bounded error \(O(e^{-2\Re T+4|\Re s|})\) (product expansion; the same error bound covers the distant images in the core log); their integrals against each nonperiodized density \(\rho(s+i v_e)\) give the coefficient, all tails/images and errors after summation costing \(O(1+N e^{-a_0\Re T/2})\) since \(2<a_0<4\). The remaining principal integral per background is the whole-line core-log convolution shifted by \(i(x_e-v_e)\). The derivative of that convolution has transform \[2\pi i\widehat\rho(p)\frac{z-z^7}{1-z^8} =2\pi i\frac{z^2+z^3-z^5-z^6}{1-z^8},\qquad z=e^{-Lp},\] obtained from the localized coth difference. Thus the convolution itself is \(\log[J(-is)/J(is)]\) at \(s=i(x_e-v_e)\): the derivative there has the same transform by shifting across a coth period, and both functions are odd. Contours can be displaced within this convolution strip by keeping the two sets of poles separated (the absolute shift in the difference is at most \(L\), individual right displacements strictly inside their strips).

Use maximum modulus after removing \(C_2 e^{-2T}+C_1e^{-T}\) by an exponential analytic multiplier. The determinant in the two-color transform adds \((3/8)\log N+O(1)\). To retain the bounded remainder, put \(\mathfrak d(p)=-\log(1-e(p))\) and write its logarithm exactly as \[2\sum_{n\ge1}\mathfrak d(2\pi n/T) =\sum_{n\in\mathbb Z}\mathfrak d(2\pi n/T)-\mathfrak d(0).\] The symbol \(\mathfrak d\) is even, regular at zero, and exponentially decaying. Its inverse transform at nonzero period translates is exponentially small, by a fixed contour displacement and bending of the tails. Periodization therefore gives \(\pi T/(16L)-\mathfrak d(0)+O(e^{-c\Re T})\), as in the two-line calculation. The two square-root factors have constant product. Since \(\Re T=(3/4)\log N+O(1)\), the real part gives the stated bound uniformly on the nome circle. Formula (67), with \(e^{I_s(0)/2}=\lambda_b/\sqrt C\), cancels the all-positive prefactors in (55) along with the site sum from (63), leaving exactly \(2 f_N^{(0)2}\) in the numerator relative to \(2 f_N^2\), where superscript \((0)\) indicates using \(h_0(v)=-\lambda_b\tan(\lambda_b v)/2\) instead of \(h\). This follows also by the Schur Pfaffian product at shifted variables as in the homogeneous calculation.

We need \[ \log\big(f_N^2/f_N^{(0)2}\big)\ge (5/24)\log N-O(1) \tag{68}\] for these rapidities uniformly. Here is a shifted comparison. Temporarily center the variables in \([-L/2,L/2]\) and make them distinct (or perturb into a slightly larger interval). The kernels by their partial fractions are integrations of \(-\sinh(vp)\) against the two positive-ray pole densities \(\nu_0,\nu\) from the determinant comparison, ratio \(R(p)=\nu/\nu_0\ge1\). Expanding a Pfaffian of size \(N=2s\) by antisymmetry therefore gives the integral of the exponential alternant on exponents \(\pm p_1,\ldots,\pm p_s\) against the corresponding product measure, up to common constants. One may cut off at 0 before taking limits by entries. This alternant has fixed sign for distinct ordered variables when exponents are laid out in pairs (pair swaps do not change sign), since exponential polynomials of \(N\) such distinct exponents have at most \(N-1\) distinct real zeros by Rolle induction. Jensen thus bounds the log Pfaffian ratio below by its linear statistic with \(g=\log R\) in the \(h_0\) ensemble, equivalently the log derivative of \(\operatorname{Pf}(h_0+t h_g)\) at zero, where \(h_g\) uses pole density \(g\nu_0\). Limits at coincidence preserve this comparison by confluent determinants.

The doubled linear statistic has the form \(A\log N+O(1)\), with a constant \(A\) independent of the rapidities and of the even size \(N\). We verify this uniform assertion before using coincidence to identify a lower bound for \(A\). Write \[H_t=\big[h_0(v_i-v_j)+t h_g(v_i-v_j)\big]_{i,j=1}^N, \qquad C(v)=\prod_j[-i\cot(\lambda_b(v-v_j))].\] If \(\Gamma\) positively encircles the distinct sites and initially no other poles, Schur’s product and its minors give the exact identity \[2\left.\frac{d}{dt}\log\frac{\mathop{\mathrm{Pf}}H_t}{\mathop{\mathrm{Pf}}H_0}\right|_{t=0} =-\frac4{(2\pi i)^2}\int_\Gamma\!\int_\Gamma C(v)C(w)h_0(v-w)h_g(v-w)\,dv\,dw.\] For the coefficient, put \(r_i=\operatorname{Res}_{v=v_i}C(v)\). Cancelling factors not incident to the two removed sites in each Schur minor gives the undoubled derivative as \(-4\sum_{i<j}r_i r_j h_0(v_i-v_j)h_g(v_i-v_j)\). The normalized double integral equals \(2\sum_{i<j}r_i r_j h_0(v_i-v_j)h_g(v_i-v_j)\); its diagonal-site residues vanish because both kernels are odd. Doubling the derivative therefore gives the displayed coefficient \(-4\), independently of the sites and of every even \(N\). The odd function \(h_g\) is bounded analytic on strict substrips \(|\Re v|<5L\), by exponential decay of \(g\nu_0\); at frequency zero the density may have a constant times \(1/p\), whose odd sine integral with cutoff remains bounded. Expand first the \(v\) contour to a rectangle of vertical edges at \(\pm2.2L\) and heights \(\pm a_0^{-1}\log N\); then \(w\) similarly, with imaginary half-height one larger. Only poles crossed are now from \(h_0\) at \(v-w=\pm b\); their contributions do not depend on the sites since \(C(v)C(v\pm b)=1\). The remaining integral is bounded uniformly, because \(|C|\) on the vertical edges is at most \(\exp(-cN e^{-a_0|\Im v|})\), on horizontal caps it is bounded, and any simple poles on the contour product itself are only transversal edge-cap intersections (locally absolutely integrable; do the deformation iteratively off these null crossing parameters). In particular the growing vertical side lengths cost only a bounded factor: \[\int_0^{a_0^{-1}\log N}e^{-cN e^{-a_0t}}\,dt =a_0^{-1}\int_1^N e^{-cx}\,\frac{dx}{x}=O(1).\] The extra unit of height for the \(w\) contour also costs only a bounded factor. For fixed \(v\) off the transversal intersections, each crossed pole \(w=v\pm b\) contributes a constant residue: the identity \(C(v)C(v\pm b)=1\) removes all site dependence. Its subsequent \(v\) integral has bounded horizontal-cap contributions and vertical pieces of length \(2a_0^{-1}\log N\). The coefficient in the double-residue formula is the same for every even \(N\), as the exact identity shows. The remaining double integral is uniformly bounded by the estimates just given. These facts prove \(A\log N+O(1)\) uniformly, and also justify the confluent limits, where the Schur formula is nondegenerate after division by its vanishing Vandermonde factors.

At coincidence, twice the linear statistic is exactly the compressed trace test from the homogeneous saturation argument, by Taylor row and column differences. That test has lower limiting coefficient \(5/24\), obtained there from Fatou’s lemma and the explicit orthogonal polynomial kernel. Hence \(A\ge5/24\). Applying the same Jensen inequality at the shifted sites proves (68) with a bounded additive error. This argument uses a lower bound for the explicit linear statistic, rather than inferring a bounded error from an exponent-level asymptotic alone.

We can now assemble the estimate. The logarithm of the normalized marked-polygon mass is at most \[-2R+\frac52(R-2u_0) +\left(\frac38-\frac5{24}\right)\log N+O(1).\] The choices \(R=(3/4)\log N+O(1)\) and \(2u_0=(3/4)\log d+O(1)\) turn this into \[-\frac43\log N+\frac{15}{8}\log(N/d)+O(1).\] Exponentiation proves Theorem [u:two-edge-theorem]. If the shorter core has only a bounded number of cells, the same proof uses the uniform edge bounds there; increasing the absolute constant covers these small values of \(d\).

Two exterior sewing estimates

Choose a triangular lattice vertex as origin and take the pure-support domains \(D=D(0;\mathbf b)\) of (53), with \(|b_j-R|\le\varepsilon R\) for a fixed sufficiently small \(0<\varepsilon\le1/20\). These are convex hexagons. Their minimum side length is then of order \(R\), and Section 8.3 supplies the uniform nesting and central-occupancy estimates for every such domain, including independent bounded row roundings.

Use a random nested family \(D_h\) indexed by \(h\) in a window of size \(cR\), \(c>0\) small. Move each nominal support outward at speed 1 from a regular arrangement plus a perturbation \(c' R f(V_i+A_i h/R)\), with \(c'>0\) small, \(f\) a unit-period triangular zigzag of absolute slope 1, \(V_i\) uniform modulo 1, \(A_i\) uniform on \([-1,1]\), independently. Given \(r<s\) in that window, conditional on the phases at \(r\) the nominal supports at \(s\) are independent with densities at most \(C/(s-r)\), since slopes’ parameters remain independent and the zigzag has boundedly many monotone branches over the allowed range.

For each facet choose a fixed nondecreasing rounding map to its admissible row levels, depending only on that facet’s nominal support and with uniformly bounded error; floor, ceiling, or nearest-level rounding with a fixed tie rule will do. Use the same map at every \(h\). Each nominal support has derivative at least \(1-c'>0\) wherever differentiable, so rounding preserves nesting. It also preserves independence between facets conditional on their inner phases. If its error is at most \(K d_0\), an atom of the rounded support is the image of an interval of length at most \(2K d_0\). Thus the conditional density bound becomes an atom bound \(C\min(1,Kd_0/(s-r))\), sufficient for all the averages below. Boundary gaps are \(O(1+s-r)\) and at least \(c_0(s-r)-O(1)\). The particular monotone rounding here constructs a nested family; the estimates for an individual domain allow every rounding satisfying the displayed support bounds.

Use complete chords in \(D_h\) with ordered boundary ends \(p,q\) binned by dyadic separation \(l\asymp |p-q|\) (of at least constant order) and by the two values \(t_i\) of one plus distance to nearest corner capped at order \(l\), also to within dyadic factors. Orientations do not affect any of our bounds. We describe two sewing estimates for a chord event of mass \(E_h\) in such a bin. Its contents such as marked vertices may be constrained arbitrarily as a deterministic event given \(h,D_h\).

Corner gains and the marked-pair sum

The previous estimate is sufficient for one endpoint bin. To sum all bins with a bounded constant, we retain the gain when an endpoint lies close to a corner. Write \(t=\min(t_1,t_2)\), with \(1\le t\le l\). The two estimates before averaging the cut gap \(g\) give the combined cost \[C(1+\min(g,t))^{-1/2}\min\{1,t/(1+g)\}.\] For each fixed inner level \(r\), the unnormalized gap law on the half-square \(r<s\) has density at most \(C/l\) over an interval of length \(O(l)\), with the analogous bound for its rounded discrete law. This uniformity allows the estimate to be integrated against the possibly nonconstant mass \(E_r\). Splitting at \(g=t\) shows that its average is at most \[\frac C l\left(\int_0^t(1+g)^{-1/2}\,dg +t(1+t)^{-1/2}\int_t^{Cl}\frac{dg}{1+g}\right) \le C_\eta l^{-1/2}(t/l)^\eta\] for any fixed \(0<\eta<1/2\). The bounded-gap atoms satisfy the same estimate after increasing \(C_\eta\).

Keeping this gain in the first sewing argument gives \[ \overline E\le C_\eta l^{3/4}(t/l)^\eta\mathcal K. \tag{71}\] Indeed the first moment in the target polygon measure is at least \(cF\overline E\), while the second is at most \(CF\overline E\,l^{-1/2}(t/l)^\eta\); use \(F\asymp l^{-5/4}\). For each smaller corner bin \(t\), the larger one has only \(O(1+\log(l/t))\) possible dyadic values. Thus their total contribution is bounded by \[C_\eta\sum_{j\ge0}(1+j)2^{-j\eta}<\infty.\] The same summation shows that bins with \(t/l\le\delta\) have total coefficient tending to zero as \(\delta\downarrow0\).

Let \(\ell_h\) denote the length of a chord in \(D_h\), and retain chords of diameter at least \(R/3\). Their mass is at most \(CR^{3/4}\). Their first-length mass is at least \(cR^{25/12}\): every path through a sufficiently central link is macroscopic, and Proposition 14 with Theorem 3 supplies mass at least \(cR^{1/12}\) at each of \(cR^2\) such links. The upper first-length mass over all chords is \(CR^{25/12}\) by the same identity.

To estimate the second moment, first omit the initial and terminal visited centers. The ordered pairs involving these at most two centers number \(O(\ell_h)\), so their integrated contribution is absorbed by the first-length bound. At each remaining center, record the directions of the two tiling edges crossed by its incident path links. Divide these centers into finitely many classes according to triangle type, lattice translation modulo two, and that pair of tiling-edge directions. The square of the total number of remaining centers is bounded by a fixed multiple of the sum of the squares of the class counts. Within a class, pairs with axial or zero displacement contribute at most \(CR\ell_h\) for each chord: a bounded domain of diameter \(CR\) has only \(CR\) positions on the finitely many axial lines through a given center. Their integrated contribution is therefore at most \(CR^{37/12}\).

For a remaining ordered pair, choose the two lattice axes spanning the strict sector containing its displacement. At least one of the two recorded tiling-edge directions is one of those axes. Since the two centers have the same triangle type, the corresponding crossed edges are translates and supply identical axial marks used by internal links of the chord. Their displacement has even coordinates in the chosen lattice basis, by the translation class modulo two. Hence \(d=P_1+P_2\) is even, as required in Theorem 2. Choose an integer period multiple \(kP\) with length comparable to a sufficiently large constant times \(R\), so the sewn polygon projects injectively. Then \(N\asymp R\) and \[\mathcal K\le C R^{-4/3}(R/d)^{15/8}.\] There are \(O(R^2)\) choices of the first center and \(O(d)\) displacements with \(P_1+P_2=d\). Consequently \[ \sum_{\text{eligible pairs}}\mathcal K \le CR^2R^{-4/3}R^{15/8}\sum_{1\le d\le CR}d^{1-15/8} \le CR^{8/3}. \tag{72}\] Apply (71) to each pair event. The corner bins sum with bounded total coefficient, and the separation factors \(l^{3/4}\) sum geometrically up to \(CR\). We obtain \[ \overline{\sum_{\gamma\text{ macroscopic in }D_h} \ell(\gamma)^2w(\gamma)}\le CR^{41/12}. \tag{73}\] The contribution from bins \(l\le\delta R\) has upper bound \(C\delta^{3/4}R^{41/12}+CR^{37/12}\). On the remaining separations, bins with \(t\le\delta'R\) likewise have arbitrarily small relative coefficient when \(\delta'\) is sufficiently small. These are estimates for actual endpoint restrictions on the chords, not an assumption that endpoints avoid corners.

A positive fraction of long bridges

We now prove (3). Write \(M_1\) for the averaged first-length mass of the macroscopic chords. We have \(M_1\ge c_1R^{25/12}\). By Cauchy–Schwarz, a class with second moment at most \(\varepsilon R^{41/12}\) contributes at most \[(CR^{3/4}\cdot\varepsilon R^{41/12})^{1/2} =C\sqrt\varepsilon\,R^{25/12}\] to \(M_1\). Choose \(\delta>0\) and then \(\delta'>0\) so that removal of close endpoints and small corner bins loses less than one quarter of \(M_1\). Chords shorter than \(aR^{4/3}\) contribute at most \(CaR^{25/12}\), which is less than another quarter if \(a>0\) is fixed sufficiently small.

The retained chords therefore have averaged first-length mass at least \(c_2R^{25/12}\). Their endpoint separation and corner distances are all bounded below by fixed multiples of \(R\), so they occupy only a bounded number of bins. In at least one bin the first-length mass has the same lower order. Applying Cauchy–Schwarz to that bin and (73) gives its averaged ordinary mass at least \[c\frac{R^{50/12}}{R^{41/12}}=cR^{3/4}.\] The second sewing estimate extends these chords to bridges between any prescribed pair of horizontal caps whose distances above and below the origin lie in \([AR,2AR]\), for a fixed sufficiently large \(A\). Since \(l\asymp R\), that estimate gives total bridge mass at least \(cR^{3/4}\) when the bottom start is free over the permitted interval of \(O(R)\) ports. Every output bridge contains the retained chord, and hence has length at least \(aR^{4/3}\).

Translation invariance makes the unrestricted mass from each bottom port equal. Removing the confinement and dividing by the number of permitted starting ports yields \[B_h\{\ell\ge aR^{4/3}\}\ge cR^{-1/4}\] for every admissible cap separation \(h d_0\in[2AR,4AR]\). For an arbitrary sufficiently large integer \(h\), choose \(R\) proportional to \(h\) so the selected caps lie in these ranges; admissible rounding changes only fixed constants. This proves (3) at every sufficiently large height.

The strip first moment and completion of the proof

It remains to prove the first-length upper bound. In the infinite strip, summing occurrences of internal links over bridges from one fixed bottom port is the same as summing the occurrence mass with free bottom port over link representatives modulo translations along the strip. There are \(O(h)\) such representatives. Proposition 14, first in finite convex truncations and then by monotone exhaustion, bounds the occurrence mass at each link by the positive enclosing-polygon partitions at its adjacent lattice points.

These partitions are at most \(Ch^{1/12}\) even though the strip is infinite. To see this, take a pure lattice period vector tilted relative to the strip’s horizontal tangent, of sufficiently large even length comparable to \(h\), with transverse projection greater than \(2h d_0\). The strip and its translate by half this period embed disjointly in the resulting cylinder. The disk enclosed by each polygon stays inside its convex strip, so it cannot contain the mark in the translated strip. Thus two independent systems enclosing corresponding points inject into a system separating the antipodal marks. The bounded-factor cylinder estimate in Section 8 is at most \(Ch^{1/6}\), so the square of the strip partition is at most that quantity.

Summing over the \(O(h)\) link representatives gives \[\sum_{\gamma\in\mathcal B_h}(\ell(\gamma)-1)w(\gamma) \le Ch^{13/12}.\] The missing one center per bridge adds only \(B_h\le Ch^{-1/4}\). This proves (2). The strip mass (1) was proved in Sections 2–3, and Proposition 11 gives (4). Theorem 1 is complete.