Critical crossing mass and related laws

Nienhuis’s analysis of the dilute \(O(n)\) model led to the honeycomb critical point and the critical-exponent predictions associated with planar self-avoiding walk [19]. Duminil-Copin and Smirnov proved the connective constant \(\sqrt{2+\sqrt2}\) by a parafermionic observable and its boundary cancellation [6]. We use that local cancellation, prove the particular finite flux identities needed here, and keep the visited-vertex convention explicit.

The sharper crossing-mass power \(h^{-1/4}\) is recorded as a prediction in [6]. Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved that the critical bridge mass tends to zero [2]. Glazman and Manolescu [13] obtained a quantitative subsequence bound, as well as invariance of boundary two-point functions in columnwise rhombic half-planes with Yang–Baxter weights. Krachun and Panagiotis proved a polynomial upper bound on the critical bridge mass [17]. These crossing-mass estimates do not by themselves control a first-length insertion. In particular, the all-length means considered here are distinct from displacement laws for uniform walks of a fixed length.

The companion paper [20] proves the same all-length bridge-mean exponent and retains it inside a fixed corridor of sufficiently large width \(Ch\). It also obtains bridge mass comparable to \(h^{-1/4}\). Our proof of the strip and hexagon results is given independently below; we do not use the companion’s mean theorem.

A polygon observable and its construction

The first-length insertion will be obtained by summing the mass of paths through a marked vertex. The corresponding algebraic observable counts surrounding polygons instead of paths. Identify horizontal translates by \(2n\), forming an infinite honeycomb cylinder, and mark two face centers on one horizontal cut, separated by \(n\) sites. Let \(F_n(\eta)\) sum over systems of pairwise vertex-disjoint unoriented simple polygons, each separating the two marks. A polygon \(P\) has weight \(\eta x_*^{|P|}\), and the empty system has weight one. Each polygon crosses the intervening marked interval, so this is a polynomial in \(\eta\); its coefficients are convergent weighted sums over polygon shapes on the infinite cylinder.

To construct \(F_n\), record at each of \(N=2n\) edges on a cylinder cut whether the edge is empty or carries either orientation of a path. The oriented pieces carry local turn phases, and adding a row acts linearly on the resulting three-state tensors. We construct a common eigenvector of eigenvalue one whose entries are polynomials in parameters attached to those edges. Dividing by its empty component gives the normalized vacuum: at the physical honeycomb parameters its entries are the amplitudes of the half-cylinder path systems. Pairing two such states with a twist between the marked faces cancels nonseparating loops and assigns each separating loop weight \(\eta=2\cosh s\).

The normalization \(F_n(0)=1\) is essential. Both the finite color formula for the numerator and the Pfaffian denominator have exponential growth in \(n^2\). The polynomial power survives only after those factors cancel in the same normalization. Uniformity for complex \(s\) also has a positive-measure consequence: the zero-free disk controls polygon incompatibilities when the cylinder is opened into the plane.

This construction uses polynomial exchange equations and reductions at special values of neighboring parameters. Di Francesco and Zinn-Justin [5] used these ideas for a multiparameter sum rule in the dense \(O(1)\) model. Garbali and Nienhuis [11, 12] developed \(q\)KZ and fusion recurrences and determinant expressions for the dilute \(O(1)\) model with open boundaries; they also discussed the polynomial construction at loop weight zero. Here the periodic twist-\(i\) state, its empty-component normalization and its physical separator pairing are constructed explicitly. The companion’s disk states represent the same homogeneous separator sum; Section 4 compares the common scalar polynomial and physical observable. The inhomogeneous vectors and normalized complex estimates here are proved separately.

The local transfer weights belong to the dilute-loop and three-state vertex-model setting of Zhou and Batchelor [24]. Related finite recurrences and corner-transfer calculations for restricted-height dilute models were developed by Warnaar, Pearce, Seaton and Nienhuis [23]; Batchelor and Yung [1] obtained integrable boundary weights for the ordinary and special transitions of honeycomb polymers. Bleher and Fokin’s six-vertex analysis with domain-wall boundaries likewise connects determinant partition functions to orthogonal-polynomial asymptotics [3]. These are methodological comparisons, not inputs for the vacuum or complex-twist estimate above.

Route through the proof

The proof first obtains two different mass estimates. After the flux and convergence bounds in Section 2, Sections 3 and 4 construct the polynomial vacuum and identify its pairing with \(F_n\). Differentiating a row at zero loop fugacity then gives the boundary mass \(b_h=h^{-1/4+o(1)}\) in Section 5. This calculation is independent of the bulk pairing asymptotics.

Section 6 proves Theorem 4. A finite color sum becomes contour integrals, with a separate Pfaffian lower bound having bounded logarithmic error. Periodizing the integrals in an auxiliary spectral direction permits a circle-determinant expansion. After that expansion, a positive Fourier decomposition controls the remaining interacting variables. Their sum stays bounded after its explicit pressure and twist factors are removed. This spectral period is distinct from the physical cylinder circumference \(2n\).

Section 7 uses the zero-free disk and six lattice rotations to localize polygons around one face: their planar partition function at radius \(R\) has exponent \(1/12\). The finite exploration identity in Section 8 converts this nesting mass into vertex-visit mass. Summing over \(h\) layers and using horizontal translation gives mixed-exit first-length mass \(h^{1+1/12+o(1)}\). In the hexagon, dividing visit mass by the free-boundary normalizer \(R^{3/4+o(1)}\) gives the visit probability \(R^{-2/3+o(1)}\) and hence the stated mean.

For the strip, the mixed mass could still be carried by returning arches. Section 9 compares intersecting bridge–arch pairs with bridge–bridge pairs through a marked tripod. A translation argument supplies sufficiently many bridge hits on each deep arch, so bridges retain the same first-length exponent. Only then do we divide by \(b_h\), obtaining \[\mathbb E_h|\gamma| =\ell_h/b_h =h^{13/12+1/4+o(1)}=h^{4/3+o(1)}.\]

Criticality of the weight.

The unfolding argument follows the bridge-decomposition method of Hammersley and Welsh [16], as used on the honeycomb lattice in [6]. We include the geometric normalization needed for our ports. A vertex-to-vertex weak bridge of positive span (intermediate heights between those of its endpoints inclusively) can be extended at start and end to bracketing layer cuts using only strictly more extreme vertices, at most one extra vertex per end. Hence by the strip upper bound the weighted sum for each span and fixed start is uniformly bounded.

Count vertex SAWs of length \(L\) from a fixed start by splitting at a lowest vertex and considering the two weak half-space walks from there. On each, cut at the successive last maximum, last minimum of the remainder, and so on. This gives weak bridges of alternating sign with decreasing positive spans (the first two can be equal), in units of one third of a layer, with sum at most \(2L\) per half; there are no zero-height steps. There are \(O(\sqrt L)\) pieces and \(\exp(O(\sqrt L\log(L+1)))\) span lists. One can enumerate the two walks relative to the splitting vertex (two translation types suffice) and then the translate back to the given start is fixed. Thus multiply the uniform bounds allowing also a bounded weight factor per joint. SAW counts have exponential rate at most \(1/x_*\).

A strictly smaller rate would make the vertex SAW susceptibility (weighted sum over all lengths at \(x_*\)) finite, hence by trimming port half-edges would make \(\sum b_h\) finite, contradicting the bridge lower bound. The rate exists, e.g. by submultiplicativity and lattice transitivity. Thus the weight \(x_*\) is critical.

A flat cylinder identifies horizontal translates by an integer \(N>0\). The following fixed-width estimate will justify the row-transfer limits. Its constants may depend on \(N\).

Polynomial vacuum

Polynomial exchange and fusion constructions occur in the dense and dilute \(O(1)\) models [5, 11, 12]. We construct the present normalized vacuum directly by polynomial interpolation and determine its empty component. Its Pfaffian formula supplies the denominator of the polygon observable.

Interpolation of the contraction

The following characterization will identify the finite color formula in Section 6.1. It applies at generic parameters; it does not assert that \(S_N\) is nonzero at every complex specialization. In the displays below we separate the site lists of the two blocks by a semicolon.

Row susceptibility at infinity

Use even widths \(N=2m\). Let \(G_N\) be the half-plane mass, from a fixed bottom port, of arches with positive endpoint displacement \(0<d<N\) whose projections to the cylinder of period \(N\) are simple. The strip flux bound gives \(G_N\le1/(2\cos(3\lambda))\). We will express this deficit through one Taylor coefficient of a row eigenvalue.

For the common eigenline near twist \(\kappa=i\), differentiate its row eigenvalue in \(\eta=\kappa+\kappa^{-1}\) at zero. Call the answer in homogeneous data \(H(\theta)\), with \(z=e^{2iv},\ \theta=2v-3\lambda\); we also write \(H(z)\) when using the transfer argument. It tends to 1 as \(z\to0,\infty\) by triangularity. It is even: at real \(v\) near \(\lambda\), empty-to-empty powers agree with those at \(3\lambda-v\) by reflection of the unoriented loop sum (\(u_1,u_2\) and \(w_1,w_2\) interchange). The simple eigenvalue strictly dominates nearby at both parameters, so they agree there on that eigenvalue and then by continuation. Its denominator may be taken to be \((\cos\lambda+\cos\theta)^N\).

Before making sites homogeneous, transfer fusion gives for the derivative as a function of transfer argument \(z\) both relations \[H(z)+H(Tz)=0,\qquad H(z)+H(t^2z)-H(tz)=0\] at every \(z=x_j\) (base eigenvalue 1). Thus the relations have zero jets of orders below \(N\) at the common site 1, by analyticity in sites and coalescence (all the denominators here regular locally; e.g. divide by \(\prod(z-x_j)\) for nearby distinct sites and bound inside a small circle). In \(\theta\) notation the site and its shifts \(t,t^2,T\) correspond to \(-3\lambda,-\lambda,\lambda,3\lambda\). By evenness the jets are \[H^{(2j)}(3\lambda)=0,\qquad H^{(2j+1)}(3\lambda)=2H^{(2j+1)}(\lambda)\quad(0\le j<m).\]

Represent the cosine-rational function as \[H(\theta)=1+\int_{\mathbb R} e^{\theta p} \frac{\sinh(\lambda p)U(p^2)+p\cosh(\lambda p)V(p^2)}{\sinh(\pi p)}\,dp\] with polynomials of degrees below \(m\). Indeed, near the real angles used here, \[\int_{\mathbb R}\frac{e^{\theta p}\sinh(\alpha p)}{\sinh(\pi p)}\,dp =\frac{\sin\alpha}{\cos\theta+\cos\alpha}.\] Its successive even \(\theta\)-derivatives and their \(\alpha\)-derivatives give precisely a basis by increasing pole order. The elementary transform follows by taking differences of \({\rm pv}\int e^{u p}/\sinh(\pi p)\,dp=\tan(u/2)\). For real \(|u|<\pi\), take the integral on \(\mathbb R+i\epsilon\), \(0<\epsilon<1\), shift that line by \(i\), and get \(-2i e^{iu}/(1+e^{iu})\) by the pole at \(i\). As \(\epsilon\downarrow0\), the line integral gives the principal value minus \(i\), from \(1/(\pi p)\) at zero.

Put \(x=\lambda p\), \(s=p^2\), and let \(\mu\) be the measure on \(s\ge0\) pushed forward from \[\frac{\cosh(3x)\sinh x}{\sinh(8x)}\,dp.\] Define the test and trial spaces \[\begin{aligned} \mathcal F_m&=\operatorname{span}\{s^j,s^j p\tanh x:0\le j<m\},\\ \mathcal G_m&=\operatorname{span}\{s^j,s^j p\coth x:0\le j<m\}. \end{aligned}\] The even functions of \(p\) here are regarded as functions of \(s\), with their continuous values at zero. The jets say precisely that \(R(s)=U(s)+p\coth x\,V(s)\in\mathcal G_m\) satisfies \[ \int_{[0,\infty)}[-R(s)]f(s)\,d\mu(s)=f(0) \qquad(f\in\mathcal F_m). \tag{2}\] This uses the identities \[\cosh3x=(2\cosh2x-1)\cosh x,\qquad \sinh3x-2\sinh x=(2\cosh2x-1)\sinh x.\] At \(z=0\) the first Taylor coefficient is \[ [z]H=2\sin\lambda\ e^{-3i\lambda}R(-1), \tag{3}\] by shifting the \(p\) contour upwards just past \(i\) at fixed \(\Re\theta=0,\ \Im\theta\to+\infty\): the pole there contributes \(2\sin\lambda\,R(-1)e^{i\theta}\) and the new integral is \(o(|z|)\) by exponential decay.

There is also a path interpretation. Take a homogeneous finite stack of honeycomb rows with one further row of parameter \(v\) above, empty boundaries, and divide by the empty-to-empty value without the further row. This tends to the row eigenvalue uniformly near twist \(i\), by strict spectral dominance and commutation. Its derivative at \(\eta=0\) counts a single essential loop using the further row. This follows by the same arc/turn embedding for angles near \(\lambda\), hence identically. With empty top the loop is either purely horizontal in that row (weight \(o^N\)) or has horizontal intervals connecting bottom ports there, using equal positive numbers of \(u_1,u_2\) and otherwise only factors \(o\) along intervals. Now \[o=\frac{(1-z)(1-t^{-3}z)}{(1-t^2z)(1-Tz)},\qquad u_1u_2=o\,\sin^2(2\lambda)/[\sin(2\lambda+v)\sin(3\lambda+v)]\sim-2 e^{5i\lambda}z .\] Thus the only other contribution to \([z]\) is a single lower arch between distinct ports plus one horizontal connection in the further row. Such an arch lifts with endpoint displacement strictly below \(N\) in absolute value (a larger displacement interlaces its translate), and of the two horizontal connections exactly one closes it with essential winding. Conversely every arch counted by \(G_N\) contributes, with this choice of starting end unique for each unoriented cylinder arc. The reflected version has the same weight. Passing to the infinite lower stack therefore gives \([z]H=N([z]o-2e^{5i\lambda}G_N)\). This passage on derivative and coefficient is also valid by spectral convergence, uniformly in the twist perturbation and in regular further-row arguments near \(z=0\). Combining \([z]o=2e^{5i\lambda}/(2\cos(3\lambda))\) with (3) gives the promised identity: \[ \frac1{2\cos(3\lambda)}-G_N=\frac{\sin\lambda}{N}[-R(-1)] . \tag{4}\]

Positive kernel estimate

By (4), it remains to prove \[ -R(-1)=N^{3/4+o(1)}. \tag{5}\] The two spaces in (2) are different, so that equation does not itself give a positive polynomial kernel. We first express its solution as a positive average of ordinary kernels over two pole sets. A comparison valid for every pole set gives the upper bound. For the lower bound, low poles occur with probability tending to one; on that event a smaller polynomial space admits the reverse comparison.

The partial fractions of \(p\tanh x\) and \(p\coth x\) are respectively \((2/\lambda)s\sum_{a}1/(s+a)\) and \(1/\lambda+(2/\lambda)s\sum_b1/(s+b)\), with \(a=(8l)^2\) at positive half-integers \(l=1/2,3/2,\ldots\), \(b=(8l)^2\) at positive integers. Equivalently the all-pole products \(\prod(1+s/a),\prod(1+s/b)\) equal \(\cosh x,\sinh x/x\). For example integrate \(f(w)/(w^2-x^2)\) for \(f=\tanh,\coth\) on expanding squares spaced away from poles (where \(f\) stays bounded); residues at \(\pm x\) sum to \(f(x)/x\), while the poles of \(f\) each have residue 1 for \(f\). The products follow by integrating logarithmic derivatives from zero.

For subsets \(A,B\) of size \(m\) of the respective pole sets, write \(\pi_A(s)=\prod_{a\in A}(1+s/a)\), similarly \(\pi_B\), and \(dw=d\mu/(\pi_A\pi_B)\). Let \(K_d^w\) denote the ordinary polynomial evaluation kernel through degree \(d-1\): if \(P_j\) are orthonormal for \(w\), then \(K_d^w(u,v)=\sum_{j<d}P_j(u)P_j(v)\).

Comparing the strip tails

The total half-plane arch weight to the right is now exactly \(1/(2\cos(3\lambda))\): this was the upper bound by flux, and \(G_N\) tends to it. Equations (4) and (5) give \[d_N:=\frac1{2\cos(3\lambda)}-G_N=N^{-1/4+o(1)}\] for even \(N\). This is the mass excluded by displacement or failure of simple cylinder projection. We first transfer its exponent to a spatial maximum, and then use a second tilted direction to force the same exponent for height.

Color formula and small-cycle gas

For the balanced homogeneous case (circumference \(2n\)) the target is the following estimate, with \(C\) uniform on each compact subset of \(|\Im s|<\pi\): \[|X_{n,n}(\kappa=e^s)|\le C\ n^{\,1/6+2\Re(s^2)/(3\pi^2)},\qquad |\Im s|<\pi ,\] The normalization is \(X_{n,n}=1\) at \(s=i\pi/2\). Together with positivity the estimate will give the matching exponent at \(s=0\) and analytic control near zero polygon fugacity. The calculation uses (i) a finite color interpolation formula converted to residues; (ii) a one-sided homogeneous normalization estimate for the Pfaffian; (iii) analytic radial periodization of the residues; upper-circle resummation creates hole variables, after which only a small defect gas around an explicit pressure remains. We bound that gas after a Gaussian count summation and recover the nonperiodized value by Cauchy at zero radial nome. The radial period \(L\) is an auxiliary spectral period, not the circumference of the physical SAW cylinder. Matrix and scalar abbreviations will be local to the computations.

Here is a rational product sum for \[\mathcal P=S_N^2 X_{n,m}/\prod_{i<j}x_i x_j\] (with the denominator meaning \(\prod_{i<j}(x_i x_j)\)). Assign labels \(a_i\in\{1,0,-1\}\), total sum zero. The type of site \(P={\rm A},{\rm B}\) has sign \(s_P=+,-\) respectively. Take single-site activities \(\lambda_0^P=1\), \(\lambda_+^P\lambda_-^P=\sqrt2-2\) and \(\lambda_+^{\rm A}/\lambda_+^{\rm B}=T\kappa\). Pair factors in order \(P\to Q\) with argument \(r=x_{\rm target}/x_{\rm source}\) and labels \(a,b\) are \[l^{PQ}_{ab}(r)=u^{PQ}_{ab}\,r^{-1-ab}\prod_{j\bmod8}(r-t^j)^{d^{PQ}_{ab}(j)} .\] The leading multipliers are \(t^{-s_P b}\) for \(a=0\); for \(a\ne0\) they are \(t^{s_Q a(1-2 b^2)}\). Writing \(e_j\) for unit arrays modulo 8 and * for index negation, the tables in order \(+,0,-\) are \[d^{AA}=\begin{pmatrix}H&J&L\\J^*&H-e_1-e_{-1}&J\\ L^*&J^*&H\end{pmatrix},\] \[H=e_2+e_3+e_5+e_6,\quad J=e_3+e_5+e_6-e_0,\quad L=e_5+e_6-e_0-e_1,\] \[d^{AB}=2\begin{pmatrix} e_5+e_6&e_5&0\\e_5&e_4&e_3\\0&e_3&e_2+e_3\end{pmatrix}.\] Obtain BB and BA by reversing both labels in AA and AB respectively. Then \(\mathcal P\) is the sum over these neutral assignments of the products of all site and pair factors.

For checking, normalize by the source sign, putting \(g=s_P a,\ h'=s_P b,\ \epsilon=s_P s_Q\). Thus the divisor array is \(D^\epsilon_{g h'}=d^{AA}_{g h'}\) or \(d^{AB}_{g h'}\) for \(\epsilon=+,-\), of total sum \(2+2gh'\). Write \(t^{U^\epsilon(g,h')}\) for the leading multiplier (\(U=-h'\) at \(g=0\), else \(\epsilon g(1-2h'^2)\)). Put \(\tau_k e_j=e_{j+k}\). The tables have \(D^\epsilon_{\epsilon h',\epsilon g}=(D^\epsilon_{g h'})^*\) and \(U^\epsilon(g,h')-U^\epsilon(\epsilon h',\epsilon g)=-\sum_j j D^\epsilon_{g h'}(j)\) modulo 8.

We check the characterization in Proposition 17. Factors are reciprocal on exchanging ends, so order does not matter. Simple poles at equal same-block sites cancel by exchanging unequal labels. At ratio \(t\) the two polar assignments \((s,-s),(0,0)\) in type \(P\) (\(s=s_P\)) cancel including activities. At ratio \(T\) only \((s,-s)\) remains; its local multiplier including activities is \(2-\sqrt2\). At ratio \(t^2\) the possible pairs \((s,0),(0,-s),(s,-s)\) coalesce to their sum with local multiplier twice the single activity. These are the two \(S^2\) reductions with the monomial denominator. To see the spectator factors explicitly put \(B_{ab,c}(r;k)=l^{PQ}_{ac}(r)l^{PQ}_{bc}(r/t^k)\). Substitution in the two tables gives \[\begin{split} B_{s,-s,c}(r;1)&=t^{3s c}B_{00,c}(r;1),\\ B_{s,-s,c}(r;3)&=T^{s c}\frac{(r-t^5)^2(r-t^6)^2}{T r^2},\\ B_{ab,c}(r;2)&=\frac{(r-t^5)^2}{tr}\ l^{PQ}_{a+b,c}(r/t) \begin{cases}1&a+b=\pm1,\\t^{3s c}&(a,b)=(s,-s).\end{cases} \end{split}\] on the allowed pairs. Indeed for a fixed target label \(c\), abbreviating rows \(D_g=D^\epsilon_{g,sc}\), one reads \(D_++\tau_1D_-=(1+\tau_1)D_0,\ D_++\tau_3D_-=2(e_5+e_6)\), and \(D_g+\tau_2D_{h'}=2e_5+\tau_1D_{g+h'}\) for \((g,h')=(+,0),(0,-),(+,-)\); compare the powers and \(U\)’s at infinity. The phases disappear by neutrality. Locally \(l^{PP}_{00}/l^{PP}_{s,-s}\) tends to \(2-\sqrt2\) at \(t\), \(l^{PP}_{s,-s}(T)=-1\), and the three factors at \(t^2\) for the indicated surviving pairs are \(2,2,-(2+\sqrt2)\). At extreme parameters a site of label \(a\) grows with degree \(N-1-a^2\) in either divergent direction by neutrality; only its zero label contributes at top order and the leading phases cancel. This gives polynomiality after restoring \(\prod_{i<j}(x_i x_j)\), the degrees and both extreme decimations.

Finally at the balanced specialization cross factors at \(T\) require the sum of matched labels nonnegative, hence zero at every match. The inter-match factors do not depend on labels: with A label \(a\) matched by \(-a\), and similarly \(b,-b\), they are \[l^{AA}_{ab}(r)^2 l^{AB}_{a,-b}(Tr) l^{AB}_{b,-a}(T/r) =r^{-4}\prod_{d\in\{2,3,5,6\}}(r-t^d)^2 ,\] as required by the cross factors in the cup reduction of \(S^2\) with the monomial denominator (here \(2D^+_{ab}+(\tau_3+\tau_{-3})D^-_{a,-b}=2H\)). Local multipliers including activities are \((2-\sqrt2)\kappa^a\). This verifies the last interpolation condition and proves the sum formula.

Now specialize to two homogeneous blocks of size \(n\). Factor out the default term (all labels \(s_P\)), which equals \[B_n=(2D_0)^{n^2}D_0^{-2n}\kappa^n,\qquad D_0=2+\sqrt2 .\] The remaining gas has objects in each type of length one (denoted E) or two (D); the total virtual length \(M\) is the same in both types. An E object changes a site label from its default \(s_P\) to zero; a D object changes it to \(-s_P\). Object length counts virtual integration variables, rather than visited vertices of a walk. Integrate each top variable \(z\) around a small positive circle surrounding 1 with measure \(dz/(2\pi i z)\); the virtual variables of an object are \(z\) alone or \((z,z/t)\) respectively. Each virtual variable carries \(h^n\) evaluated there, and distinct objects carry all pairwise interactions between their virtual variables, as follows: \[h(r)=\frac{(r+t)^2}{(r-1)(r-t^2)},\quad K_s(r)=\frac{(r-1)^2(1+r^2)}{t r(r-t)(r-1/t)},\quad K_x(r)=\frac{t r(1+r)^2}{(1+\sqrt2 r+r^2)^2}\] (pairwise argument = ratio, same and different colors respectively). Sum with inverse factorials of the numbers of indistinguishable objects and activities \[A_P=\frac{1}{a_0\lambda^P_{s_P}},\quad a_0=\frac{t^2}{1-t^2},\qquad B_P=-A_P^2\,\mathop{\rm Res}_{r=t}\big(K_s(r)/r\big)\] for E,D respectively. This gives \(\mathcal P/B_n\).

For the residue check first separate base sites slightly, preserving equal block sizes and dividing by their corresponding default term, using instead of \(h^n\) products at virtual/site ratios of \(p_s(r)=t^2 r/[(r-1)(r-t^2)]\) on same-color sites and \(p_x(r)=t^6(r-t^5)^2/r\) on the others. Objects now select distinct sites of their color (coincident tops have same-color zero of order 2,1,2 for EE,ED,DD in the combined kernels). Length \(i\) means new label \(s_P(1-i)\). By the pair tables, against a still-default site the change ratio equals the string of \(p\)’s times \(t^{3s_Ps_Q 1_{i=2}}\). Between two changed sites the correction (on top of their two single-site changes) is the string of \(K\)’s times \(t^{-3s_Ps_Q(i 1_{j=2}+j 1_{i=2})}\); these identities follow just by multiplying linear factors. The constants thus give net \(t^3\) per D: for a length-2 site in type \(P\) the assigned exponent is \(3s_P\sum s_Q(1-j)\) over other sites of lengths \(j\) (taking \(j=0\) if unchanged), which equals 3 by neutrality. The extra self-site residue factors are \(a_0\), or \(a_0 p_s(1/t)\) for a D, consistent with \(B_P=t^3\lambda^P_{-s_P}/(\lambda^P_{s_P} a_0 p_s(1/t))\). No other poles intervene on these small contours. This proves the integral formula (one can truncate to \(M\le 2n\)).

For instance use the ratio (top at type \(P\))/(top at \(Q\)), with \(\epsilon=s_Ps_Q,\ g=1-j,\ h'=\epsilon(1-i)\) normalized now at source \(Q\). The correction divisor read from the tables is \[D^\epsilon_{g h'}+D^\epsilon_{1,\epsilon}-D^\epsilon_{1,h'}-D^\epsilon_{g,\epsilon} =\sum_{u=0}^{i-1}\sum_{v=0}^{j-1}\tau_{u-v} K_\epsilon^\#,\qquad (K_+^\#,K_-^\#)=(2e_0+e_2+e_6-e_1-e_7,\ 2e_4-2e_3-2e_5),\] and the analogous difference of four \(U\)’s modulo 8 is \(\epsilon[-ij+ij(j-i)/2-3(i1_{j=2}+j1_{i=2})]\), giving the stated phase including the shifts in the kernel string.

Homogeneous Pfaffian normalization

The denominator is now fixed. We return to the numerator integral and deform it without changing its physical value at zero radial nome.

Radial periodization of the small-cycle integral

Write \(\alpha=\pi/4,\ \kappa=e^s\), and use coordinates \(x=\log(z/t)\), periods \(L,2\pi i\), radial nome \(\mathfrak q=e^{-L/2}\), and \(P=\mathfrak q^2\). Define \[S(x)=2\sinh(x/2)\prod_{a\ge1}(1-P^a e^x)(1-P^a e^{-x}).\] Use kernels \(\prod_j S(x-i\alpha j)^{d_j}\) specified by these divisors (actual integer shifts, so \(4,-4\) denote two factors): \[h:\ d_{\pm1}=-1,\ d_{\pm4}=1;\quad K_s:\ d_0=2,\ d_{\pm1}=-1,\ d_{\pm2}=1;\quad K_x:\ d_{\pm3}=-2,\ d_{\pm4}=1.\] They are for the external field in \(x\) and for pair differences respectively. At \(P=0\) these agree with the small-cycle kernels with multipliers \(t^{-1},t\) stripped from the pair kernels; these phases would contribute \(t^{M+\#D}\) in total. Set \(C_L^2=\lim_{x\to0}K_s(x)/x^2\) (square root positive at \(P=0\)), \(k_L={\rm Res}_{x=i\alpha}K_s(x)\). Use E activity \(a_L=-i C_L e^{-s/2}\) in each type and D activity \(-k_L a_L^2\) (incorporating the phases at \(P=0\); indeed \(C_\infty^2=D_0\) and \(t A_A A_B=-D_0/\kappa\) there). Multiply the integrand by \[\Theta(C)^2,\quad C=\exp(\sum_\nu q_\nu x_\nu),\quad \Theta(C)=\sum_{b\in\mathbb Z} e^{-Lb^2/2} C^b\] where \(\nu\) ranges over virtual variables including both constituents of D, and \(q_\nu\) is their color sign. D constituents are now \(x,x-i\alpha\). Integrands are elliptic in each object’s top coordinate, by neutrality and the shift formula \(S(x+L)=-e^{x+L/2}S(x)\) (the multiplier of the pair kernels moving one object of charge \(p_i\) is \(C^{-2p_i}\exp(-p_i^2L)\), canceled by \(\Theta(C)^2\)).

Denote by \(\mathcal N_n(\mathfrak q,s)\) the resulting small-cycle gas around tops \(x=-i\alpha\), summed over \(0\le M\le2n\) with its factorial measures and deformed activities, and multiplied by \(B_n\). This includes the factor \(\kappa^n\) in \(B_n\). The numerator \(\mathcal N_n\) is analytic for small \(\mathfrak q\); at zero nome the homogeneous monomial denominator in \(\mathcal P\) is one, so \[\begin{aligned} \mathcal N_n(0,s)&=\mathcal P(1,\ldots,1;\kappa=e^s)\\ &=S_{2n}(1,\ldots,1)^2 X_{n,n}(\kappa=e^s). \end{aligned}\] Even if the small-cycle sum is extended, any fixed \(M>2n\) contributes zero: resolve \(h(x)^n\) into a product with slightly displaced fields and use distinct top poles as before, then pass to the limit.

Here a single sufficiently small contour size and disk of analyticity in \(\mathfrak q\) work independently of \(n\). For top differences near zero the same-color string products have orders \(2,1,2\) for EE, ED, DD, and the opposite-color ones have no poles; other real-period images stay away. So after splitting the external poles each color selects distinct top poles (at most \(n\), length at most 2 each). Products defining \(S,\Theta\) converge normally where needed, and \(C_L,k_L\) are analytic deformations of their values at zero radial nome.

We describe term transformations first at fixed \(M\), large real \(L\). Write a radial cycle of shift \(\theta\) as \(x=r-i\alpha\theta,\ -L/2<r<L/2\), positive towards the right. Expand each small cycle as lower minus upper (shifts \(1\pm\epsilon_0\), tiny \(\epsilon_0\)). Move upper E’s up to shift 0. The pole crossed against each still-lower E of the same color is simple, producing a two-string with coefficient \(-k_L a_L^2\) (minus sign of the moving cycle). These cancel all upper D contours, which have opposite coefficient. More explicitly one can move the E contours successively; merged pairs have top near shift 0 and can move to shift \(1-\epsilon_0\). They have no further poles against remaining E or D’s in this step (same-color nearby poles of string products cancel against zeros), hence the binomial expansion over disjoint mergings gives the stated cancellation with the factorials. Cycles can be infinitesimally separated during moves. Move both lower contours together to shifts \(2+\ell\) for remaining E (now called \(\mathrm L\)) and \(2+d\) at D tops, with \(\ell=1/2, d=-1/8\). No pole is crossed (opposite-color pair distances from the upper contour remain below 3; lower top relative shifts stay smaller than 1). The other E’s are now at shift 0 (\(\mathrm U\)) with reversed sign.

In this deformation the small-cycle expansion holds by periodicity since all top contours in the initial small band have no pair poles. For each choice of \(r'\) disjoint mergings in one color, the two E factorials after summing matchings leave factorials for the two residual E counts and an extra \(1/r'!\). The residue equation is \(x_{\rm moving}=x_{\rm lower}+i\alpha\), so after moving the created contours next to the upper D contours, a fixed total \(d'\) there has relative coefficient \(\sum_{r'=0}^{d'}(-1)^{d'-r'}/[r'!(d'-r')!]\) times the D activities. Hence only \(d'=0\) remains.

For instance an E approaching a D with their virtual shift differences at \((1,2)\), \((-1,0)\) or \((0,1)\) sees no same-color pole in the product; two D’s likewise have no pole for top differences \(0,\pm1\). Newly created tops can go from \(\epsilon_0\) to \(1-\epsilon_0\) with bottoms strictly beyond 1. At the last move take linear interpolation of both lower top shifts: no U–lower constituent distance reaches 3 or recrosses 1, and the L,D top relative shift has size \(<1\).

Here are explicit zero and nonzero mode conventions. Put \(\varepsilon=2\pi\alpha/L,\ \phi=2\pi r/L,\ y=e^{i\phi}\). For a pair of constituents with kernel divisor \(d_j\), shift difference \(\theta\) (target minus source), \(p=q_{\rm source}q_{\rm target}=\sum d_j/2\), use \[W(v)=\sum_j d_j\frac{\exp[v((j+\theta)\bmod8-4)]}{2\sinh4v} =\frac{p}{4v}+W^{[0]}+W^{[1]}v+O(v^2),\quad k=\sum_j d_j(1/2+\lfloor(j+\theta)/8\rfloor)\] taking half averages of the two branches at a wall. For whole objects sum the arrays over constituents, obtaining \(W_{ij}\) when source and target are lower objects (also use copies at identical data for \(i=j\)), a two-color column \(J_i^+\) from \(\mathrm U\) to lower \(i\) (reversed = \(J_i^-\)), a symmetric matrix \(A\) between U variables, and \(H\mathbf1\) (for U) or \(h_i^+\) (for lower, reversed = \(h_i^-\)) from the external field once (\(p=0\)). In particular \(H=(1-\cosh3v)/\sinh4v\), \(A_{ss},A_{sx}\) (diagonal and off-diagonal) equal \((\cosh4v-\cosh3v+\cosh2v)/\sinh4v,(1-2\cosh v)/\sinh4v\), both eigenvalues of \(A\) positive.

These conventions follow from expanding the Gaussian-modified kernel (multiply by \(\exp[-p(x_j-x_i)^2/L]\)). It has winding phase \((-y_j/y_i)^k\), mean log \(p L/6+p\alpha^2\theta^2/L-\varepsilon W^{[1]}\) apart from phase, and log coefficient \(-W(m\varepsilon)/m\) at \((y_j/y_i)^m\), \(m>0\), with reversed shifts at negative frequency. Winding powers here can be grouped to integer powers for whole objects (off the walls this follows from even total divisor order per kernel; at the final shifts the same-contour EE walls have even order, and for DD the \(+1,-1\) differences of constituents enter together along with the two zero differences). One direct derivation uses, for each shifted sine at \(r-i\alpha u\), \[S=\mathrm{sgn}(r)e^{(|r|-i\alpha u\,\mathrm{sgn}(r))/2} \prod_{l\in\mathbb Z}(1-e^{-|r-lL|} e^{i\alpha u\,\mathrm{sgn}(r-lL)}) \qquad (|r|\le L/2).\] Signs cancel; the log jump at 0 after summing and Gaussian compensation is \(-2\pi i k\), giving the indicated phase. The mean of the product log per shifted factor is \(-2\pi^2 B_2(\{u/8\})/L\), \(B_2(w)=w^2-w+1/6\). At nonzero frequency expand the logs; integrating on the two half-lines gives sums of \(1/(a^2+\omega^2)\) against \(\cos(a\alpha u)\) and \((\omega/a)\sin(a\alpha u)\), \(\omega=2\pi m/L\). Combining with the exponential and winding terms yields the formula for \(W\) (equivalently use the Fourier series of \(e^{\omega(x-\pi)}/(2\sinh\pi\omega)\) on \(0<x<2\pi\)). At zeros take log-moduli singularities by radial limits.

In detail, for \(\beta=\alpha u\bmod 2\pi\), per unit divisor the positive log coefficient after winding removal is \(-1/L\) times \[\frac1{\omega^2}+\frac{\beta-\pi}{\omega} +2\sum_{a\ge1}\frac{\cos(a\beta)+(\omega/a)\sin(a\beta)}{a^2+\omega^2} =\frac{2\pi}{\omega}\frac{e^{\omega(\beta-\pi)}}{2\sinh(\pi\omega)}\] (average endpoints), using \(i(\phi-\pi\,\mathrm{sgn}(r))\) as the two-half log of \(-y\) and symmetry of the shifts to distribute the Gaussian terms.

Poisson-transform the two theta factors (dual indices \(b_1,b_2\), sum \(b\)). Let \(p_i=\sum_{\nu\in i}q_\nu\), \(z_0=\sum_\nu q_\nu\theta_\nu\). Let \(u\) be the column of U counts (length \(M\) minus lower virtual counts in each type); let \(e\) be the counts of lower objects (D counted once). Apart from activities, measures, powers \(e^{-sM}\), and lower-object self corrections specified below, the scalar mean is \(2\pi/L\) times \[\exp\left\{-\frac{\varepsilon}2\left[ (z_0+4b)^2/4+4(b_1-b_2)^2 +[v]\{u^t A u+2u^t(nH\mathbf1+\sum J_i^+) +\sum_{ij} W_{ij}+2n\sum h_i^+\}\right]\right\},\] where \([v]\) takes the coefficient of \(v^1\) at zero. Indeed the raw-pair Gaussian compensation sums to \((-(\sum q_\nu x_\nu)^2+\#D\,\alpha^2)/L\) by neutrality, the square canceling the modular exponential. Pair mean offsets sum to \(\alpha^2(-z_0^2-\#D)/L\) (the internal D pairs are omitted). The \([v]\) terms use the full ordered square of all objects and require diagonal corrections below. Phases outside the upper expectation are \[\prod_i y_i^{-b p_i}(-y_i)^{u_{c(i)}-n |p_i|} \prod_{i<j}(-y_j/y_i)^{k_{ij}}\] where \(c(i)\) denotes color. Upper means and smooth self factors convert upper cycles exactly to independent circular unitary measures in each color (Vandermonde modulus squared with normalized Haar circles and factorials). The inserted winding power on each upper \(y\) is \(-e_{c}-bq_c\). Additionally insert, in the exponent for each frequency \(m>0,\ v=m\varepsilon\), with \(T_\pm\) the columns of sums of upper \(y^{\pm m}\), the quadratic term \[m^{-1}\big[T_-^t(1-A)T_+ -T_-^t(nH\mathbf1+\sum_i J_i^+y_i^m) -(nH\mathbf1+\sum_i J_i^- y_i^{-m})^t T_+\big].\] The remaining nonzero-mode log is \[-m^{-1}\big[\sum_{ij}W_{ij} y_j^m y_i^{-m} -\sum_i W_{ii}(+\infty)(1-e^{-v}) +n\sum_i(h_i^+y_i^m+h_i^- y_i^{-m})\big]\] per mode.

Indeed from U to a lower object the winding \(k\) is 1 in the same color and 0 in the other, and for its external field is \(-|p_i|\). Fourier sums with coincident cycles are understood away from coincidences or with the pure log singularities factored. Including self terms requires the factor \(\exp(\Gamma_i)\) per lower object (note \(W_{ii}^{[0]}=0\) by reversal symmetry), where \[\Gamma_i=-p_i^2 L/12+\varepsilon W_{ii}^{[1]}/2+ \sum_{m>0}[W_{ii}(m\varepsilon)-W_{ii}(+\infty)(1-e^{-m\varepsilon})]/m=O(1)\] uniformly at large \(\Re L\), bounded \(\Im L\) (subtract \(p_i^2/(4m\varepsilon)\) in the series to cancel the linear growth). At U instead subtract 1 from the same-color coefficient at every frequency for the Haar separation: the resulting self log correction is exactly \(-\log(LC_L/(2\pi))\) by the limiting same-color kernel divided by \(|1-e^{i\phi}|^2\). Thus it cancels the signed measure conversion with \(a_L\), modulo the length fugacity already counted. Lower objects still have factorial measures and activities with \(e^{-sM}\) removed. These fixed-\(M\) identities continue to complex \(L\) (radial cycles along the period) near large positive real parts with bounded imaginary parts.

In this bookkeeping \(\sum_{i<j}p_i p_j L/6=-\sum_i p_i^2 L/12\) over distinct whole objects (including U). Explicitly for real period the U same-color compensated kernel at shift zero divided by \(|1-e^{i\phi}|^2\) tends at coincidence to \((LC_L/(2\pi))^2\). Its logged limit by the mean and Fourier coefficients above is \[L/6-\varepsilon A_{ss}^{[1]}-2\sum_{m>0}(A_{ss}(m\varepsilon)-1)/m .\] Negative half of this is the self correction for writing the smooth residual U interactions with both indices independently summed. The kernel at this alignment is positive on the real cycle away from its zeros, so there is no extra phase in the conversion.

Upper-edge resummation

The upper contours still contain an arbitrarily large number of points. The following circle determinant expansion sums them first and replaces them by holes on two separated contours. The dependence of its sign on the winding and circle size must be retained: it is what later permits the Gaussian sum over the original count \(M\).

The zero-winding case is closely related to the Toeplitz–Fredholm identity of Borodin and Okounkov [4]. The calculation below retains the winding and its size-dependent sign; its analytic interchange with the Hubbard fields is justified separately.

Here is the winding version of the expansion. Take potentials \(V_+(y),V_-(y)\) with strictly positive and negative powers respectively, analytic on a neighborhood of a closed annulus spanning the unit circle. For a circular unitary expectation of size \(u\) insert winding \(y^{-d}\) and \(\exp(V_++V_-)\) on each variable. Expand in additional points called holes, with signs \(\tau=+1\) (outer circle \(>1\), O) or \(-1\) (inner \(<1\), I), counts \(r,l\) with \(r-l=d\). Use normalized positive circle measures divided by factorials. The scalar prefactor is \[(-1)^{ud}\exp\sum_{m>0}m [y^m]V_+ [y^{-m}]V_- .\] Hole factors are \[\prod Y_h^{-u\tau_h}\exp\sum_h\tau_h(V_-(Y_h)-V_+(Y_h))\ \prod_{h<h'} E(Y_h,Y_{h'})^{\tau_h\tau_{h'}},\qquad E(a,b)=(a-b)^2/(ab)\] times a sign depending on \(r,l,d\) only (not \(u\)); only this independence is needed. Circles can be taken wherever the series potentials converge in the indicated annulus.

We give proof details to specify both convergence and the dependence of phases. It suffices first to use polynomial positive and negative times, as formal series. For \(d\ge0\), expand the two exponentials against Vandermonde determinants by minors and integrate: one gets \(\sum_\lambda s_{\lambda+(d^u)}(V_+)s_\lambda(V_-)\) with \(\ell(\lambda)\le u\). Here \(s_\lambda(t)=\det[h_{\lambda_i-i+j}]\), \(h(z)=\exp(\sum_{k>0}t_k z^k)\) for the respective coefficient list. Transposing partitions gives \((-1)^{ud}\sum_{\nu_1\le u}s_{(u^d,\nu)}(-V_+)s_\nu(-V_-)\). Indeed the partition determinant is a Toeplitz minor on columns \(u-i\), rows \(u-i+\lambda_i\); the complementary inverse minor on a sufficiently large finite index interval starting at 0 uses the complement \(u+j-1-\lambda^t_j\) and \(h^{-1}\), giving the transpose and sign.

Explicitly \(\det[y_j^{u-i}]\prod_j h(y_j)\) expands by matrix multiplication into determinants with rows \(y_j^{u-i+\mu_i}\) for partitions \(\mu\) (padded to size \(u\)), with precisely these minor coefficients. The reciprocal-variable determinant expands likewise. Their product integrates with \(\prod_j y_j^{-d}\) and the circle factorial measures by orthogonality of monomials, requiring positive-list indices to be the negative-list indices shifted up by \(d\); this gives the asserted first sum. In the transposed sum \((u^d,\nu)\) means prepend \(d\) parts of size \(u\).

The charged-wedge calculation is the classical boson–fermion correspondence [9]; see the Clifford-operator and Schur-function treatment in [10]. We give its coefficient calculation to fix the charge convention and the projection onto permitted occupied levels, before taking any analytic specialization.

Use wedges with decreasing occupied positions \(c+\lambda_i-i\), charge \(c\), vacancy of the partition denoted \(|c\rangle\). Insert/remove level \(j\) by \(\psi_j,\psi^*_j\) (sign by exterior order). The currents \(A_k=\sum_j\psi_{j-k}\psi_j^*\ (k\ne0)\) satisfy \[[A_k,\psi_j]=\psi_{j-k},\quad [A_k,\psi^*_j]=-\psi^*_{j+k},\quad [A_k,A_h]=k\delta_{h,-k}.\] Indeed terms act finitely on each state, and the last bracket commutes with all insertions/removals so is scalar (commute first with occupation projectors); evaluate on the vacancy, using the \(k\) possible moves up by \(k>0\). The partition amplitudes of \(\Gamma_-(t)|c\rangle\) with \(\Gamma_\pm(t)=\exp(\sum_{k>0}t_k A_{\pm k})\) are the minors above by exterior multiplication with the positive shift generating function \(h\) (can truncate the sea deep below all possible movements at any fixed weighted degree). Thus after removing the transpose sign our sum is \[\langle d|\Gamma_+(-V_+)\psi_{u+d-1}\cdots\psi_u \prod_{j\ge u+d}(1-\psi_j\psi_j^*)\Gamma_-(-V_-)|0\rangle .\] Expand the projectors (finitely effective in each weighted degree) and use generating currents \(\psi(w)=\sum\psi_k w^k,\ \psi^*(v)=\sum\psi_k^* v^{-k}\), with creations outside, annihilations inside. Commuting the \(\Gamma\)’s past the currents gives precisely the exponential factors stated. The vacancy expectation with \(r=d+l\) creations and \(l\) annihilations (ordered separately, signs from reordering depending only on counts) is up to sign the determinant with columns \(w_i^a\ (0\le a<d), v_j/(w_i-v_j)\), by summing over distinct removed negative levels and creating these plus \(0,\ldots,d-1\). The mode extractions, symmetrized using antisymmetry and summed by minors over the projection levels, give a second determinant with columns \(w_i^{-a},(v_j/w_i)^d/(1-v_j/w_i)\), apart from \(\prod_h Y_h^{-u\tau_h}\) and inverse factorials. Indeed the fixed creations use levels \(u+a\), and the \(l\) projector pairs distinct levels \(u+d+n',\ n'\ge0\). After the \(Y_h^{-u\tau_h}\) extraction, antisymmetrization in the \(w\)’s gives columns \(w_i^{-a},w_i^{-d-n'}\), in the \(v\)’s the minor on powers \(d+n'\). Sum over subsets of the \(n'\)’s (listed in order) by determinant multiplication using \(|v_j|<|w_i|\); the antisymmetrizations divide by \(r!\,l!\), with ordering signs depending only on counts. Subtract polynomial columns; the product of the two Cauchy–Vandermonde determinants gives exactly the pair factors up to sign independent of \(u\). All mode sums at fixed counts converge on the separated radii. The series thus obtained converges absolutely for deterministic analytic potentials (Hadamard bounds \(r^r C^{r}\) for the two determinants at fixed \(d\), absorbed by the factorials), giving the analytic identity. Negative \(d\) follows by inverting circles and interchanging the signs.

To elaborate, in the second determinant use the polynomial columns and the finite geometric sum so its non-polynomial columns become \(1/(1-v_j/w_i)\); thus it is a Cauchy–Vandermonde in the reciprocal variables. The product with the first determinant has Vandermonde squares divided by \(\prod(w_i-v_j)^2\), and the remaining powers per \(w_i,v_j\) are respectively \(l-r+1,r-l+1\), exactly as in the \(E\) factors. Projector subset signs, antisymmetrization signs and Cauchy–Vandermonde signs here do not depend on \(u\). For finitely many times the absolute bound above is uniform on their compact sets; Taylor coefficients agree formally (only finitely many projector subsets can act at fixed weighted degree before commuting). Then approximate analytic potentials by truncations on separated circles.

Now use Hubbard linearization of the upper nonzero modes in the preceding integral. Write \(B^\pm=nH\mathbf1+\sum_i J_i^\pm y_i^{\pm m}\) at positive frequency \(m\). In the color basis jointly use coefficient columns \[V_{\pm m}=-B^\mp/m+\sqrt{1-A}\,\xi_{\pm,m}/\sqrt m ,\] where the columns \(\xi_{+,m}=\overline{\xi_{-,m}}\) are independent standard circular complex Gaussians across modes (matrix square root in the fixed real orthogonal eigenbasis). Their average reproduces the quadratic term. Apply the winding expansion in each color, with its upper size and its winding \(d_c=e_c+b q_c\) (distinct from lower contour shift notation). Set \(Y_h=e^{\varepsilon h}y_h\), where the shift \(h\) is initially tiny positive on O, negative on I. Let \(c^\pm\) collect \(\sum \tau Y_h^{\pm m}\) by color. Averaging the exponential factors from the expansion gives exactly \[\det A(v)^{-1}\exp\left( m^{-1} \{(B^--c^-)^t A(v)^{-1}(B^++c^+)+ (c^-)^t c^+\}\right)\] per mode. Hole pair factors \(E^{\tau\tau'}\) apply within each color. Their winding/positive-modulus separation uses mean log \(\varepsilon|h-h'|\tau\tau'\), phase power \(-\mathrm{sgn}(h-h')\tau\tau'\) on \(y_{h'}/y_h\) (up to fixed signs). Thus the resulting nonzero Fourier hole-hole kernel, including the displayed Gaussian terms, is \(\tau\tau'e^{(h'-h)v}(A^{-1}+\mathrm{sgn}(h-h')1)\), with self subtractions for the original circle pair singularities. Hole-lower kernel in forward direction is \(\tau e^{-hv} A^{-1}J_i^+\), in reverse \(-\tau (J_i^-)^t A^{-1} e^{hv}\). The two factors \((-1)^{u_cd_c}\) combine with the preceding lower winding powers to leave no count sign depending on \(M\) since \(u=M\mathbf1-\mathrm{(lower\ virtual\ counts)}\).

For the Gaussian/series interchange at fixed \(L\) real and fixed discrete data before introducing holes, the random potentials are analytic on a small annulus (exponentially decaying high modes). Moreover the series is integrable absolutely. Choose shifts \(\pm h_0\) sufficiently tiny inside this annulus. Integrating the modulus against the Gaussians uses the positive stiffness \(A(v)\) after the Szegő prefactor. Apart from one-body terms (bounded per hole at fixed data), the absolute real nonzero-mode Hessian after including Cauchy factors and Gaussian gains, conjugated by charge signs \(\tau\), is, per color eigenvalue \(a\) of \(A\), \[e^{-|h-h'|v}-(a^{-1}-1)_+\sinh(hv)\sinh(h'v) -(1-a^{-1})_+\cosh(hv)\cosh(h'v).\] Indeed the linear real Gaussian sources from holes have coefficients proportional to \(2\sinh(hv)\Re(\xi y^m)\) for a real-coupled mode, \(2\cosh(hv)\Im(\xi y^m)\) for an imaginary-coupled one. The matrix on the two shift classes is positive definite when \(\tanh(h_0v)<a<\coth(h_0v)\). This holds for tiny fixed \(h_0>0\) by the explicit \(A\) (eigenvalues asymptotic to \((9/8)v,1/(2v)\) at zero and \(1+O(e^{-v})\) at infinity). At fixed \(\varepsilon\), subtract \((1-e^{-cv})1\) for very small \(c>0\), leaving positive smooth Fourier forms whose missing diagonal terms cost at most constants per hole. The subtracted kernels have pairwise gain at most \(O(c\varepsilon)\) per pair (sum the logarithms), which is absorbed by the favorable Cauchy mean \(-2\varepsilon h_0\sum_c r_c l_c\) up to constants since count differences are fixed. The factorial measures now ensure absolute convergence.

Here the small subtraction is the identity on both color and shift indices. Its distinct-point terms only concern identical shifts in one color (hence equal charge signs), with pair gain bounded by \(2\log\sup_{|z|=1}|(1-z)/(1-e^{-c\varepsilon}z)|\le c\varepsilon\). It preserves positivity if \(c\) is small: at high modes the Hessian tends exponentially to identity, and on the rest \(v\ge\varepsilon\) use strict positivity. Completing the residual (positive) ordered sums with their diagonals then only pays \(\sum_{m>0}e^{-cm\varepsilon}/m\) per hole: the Gaussian gains already have their diagonals, and the original missing Cauchy diagonal coefficient 1 was split into \(1-e^{-cv}\) (kept on distinct points only) and \(e^{-cv}\). Also \(\binom{r_c}{2}+\binom{l_c}{2}\le r_c l_c+O(1)\) at fixed count difference for the two same-shift classes in each color (the subtraction constant can be chosen below \(2h_0\)). Gaussian integrations can also be truncated in modes first (for each fixed number of points the high coefficients are exponentially small, giving integrable domination of that tail).

The absolute bounds work uniformly over the lower radial positions here at fixed preceding counts and indices. With Gaussian cutoffs the high-mode gain terms in the real Hessian simply drop out, preserving the bound, and the Gaussian cross terms with the deterministic sources are bounded by a constant per hole by high-mode decay on the tiny annulus. For each fixed number of holes the Gaussian formula passes to the infinite product as well (summable coefficient tails; one can take absolute moments of order slightly above 1 using strict Gaussian stiffness).

The residual kernels

The circle determinant has eliminated the \(\mathrm U\) variables. We first carry out the remaining transformations for a fixed nonnegative integer \(M\) and a large real period \(L\). Here \(M\) is the common total number of virtual integration variables per color in the original small-cycle gas, including variables assigned to lower objects. It is unrelated to the visited-vertex length of a walk. If \(g_c\) is the number of lower virtual variables of color \(c\), the eliminated upper ensemble had size \(u_c=M-g_c\).

We now give the remaining interactions as a matrix on a fixed list of species. The exact one-object measures and factorials are retained, together with the implicit constant phase signs. The one-object prefactors obey the stated uniform bounds and are independent of \(M\) at fixed defect data; the implicit phases have modulus one and depend only on the discrete configuration. The position-dependent phases remain explicit below. Section 6.7 will justify absolute summability of the assembled expression, complex-period continuation, and the subsequent sum over \(M\).

The hole sign \(\tau=1,-1\) refers to the outer or inner contour of an eliminated upper ensemble. Both signs occur in each gas color, whose sign is \(q\). In the moves below we continue the Fourier coefficients and their winding powers as functions of the offsets, re-expanding a pure singular factor when its Fourier branch changes. The upper circles have already been integrated out.

After integrating the Hubbard fields move hole contours O to \(h=3/8\); then move I to \(h=-1/4\). In the second move the pole with a D of the same color creates an object \(\mathrm P=(\mathrm D,\mathrm I)\) at equality of their radial variables, with \(d=h\) (here \(2+d\) is the lower top shift of D). Put free D at \(d=1/8\) and P at \(d=h=0\). Their unshifted radial variables within the period are denoted \(x_i\) in what follows (so \(y_i=\exp(2\pi i x_i/L)=e^{i\phi_i}\), and e.g. a hole has \(Y=e^{\varepsilon h}y_i\)).

Use \({\bf 1}=(1,1)^t, Q=(1,-1)^t\), and unit columns by color. Let \(t_i=\tau_i\) times the corresponding unit column for holes, otherwise zero. For a lower gas object write \(p_i\) for its signed length, \(z_i=\sum q_\nu\theta_\nu\) for its virtual angular moment, \(\lambda_i\) for its color unit column; all three are zero on holes. Use addition of data for P. The following table collects the resulting five species in either color of sign \(q\). Here \(a_i\) means respectively \(\ell,d,h,h,d=h\), the jump charge is \(j_i={\bf1}^t(\lambda_i-t_i)\), and the external-drive coefficient is \(\xi_i=|p_i|-j_i\). \[\begin{array}{c|l|rrrr} \text{species}&\text{constituents}&a_i&p_i/q&j_i&\xi_i\\\hline \mathrm L&\text{one lower variable}&1/2&1&1&0\\ \mathrm D&\text{two lower variables}&1/8&2&1&1\\ \mathrm O&\text{outer hole}&3/8&0&-1&1\\ \mathrm I&\text{inner hole}&-1/4&0&1&-1\\ \mathrm P&\text{bound D and I}&0&2&2&0 \end{array}\] For fixed species, counts and positions, we will isolate all \(M\)-dependence in a Gaussian count factor. We first define its center and the residual interactions, state the resulting factorization, and then verify the contour and scalar bookkeeping.

Continue \(J_{\rm D}^\pm\) by retaining the initial branches (coefficient dependence \(e^{\pm d v}\)). Define columns \[\begin{gathered} F_i^+=J_i^++t_i e^{h_i v}=T_i/v+f_i+r_i v+\cdots,\\ F_i^-=J_i^- -t_i e^{-h_i v}=-F_i^+(-v),\qquad T_i=Qp_i/4,\quad f_i=Qz_i/4-\lambda_i+t_i. \end{gathered}\] (terms only on applicable constituents). The dressed oriented Fourier kernel \(G(v)\) is \[G_{ij}=W^{g}_{ij}-(F_i^-)^t A^{-1}F_j^+ +t_i^t t_j\,\operatorname{sgn}(h_i-h_j)e^{(h_j-h_i)v}+C_{ij}.\] Here \(W^g\) is just the preceding bare \(W\) between lower gas parts, with actual lower-lower shift branches. The crossing correction for each I,D pair with relative offsets \(h,d\) satisfying \(d>h\), writing \(R=1\) on equal colors and \(R=-2\) otherwise, is \[C_{\rm ID}=R e^{(d-h)v},\qquad C_{\rm DI}=-R e^{(h-d)v}.\] Use half at equality, zero for \(d<h\). Expressions on constituent pairs are added. Use the winding \(k_{ij}\) from the direct interactions: gas winding already defined, plus \(-t_i^t t_j\,\operatorname{sgn}(h_i-h_j)\), plus \(-R\) from I to D in the crossing correction (antisymmetric, half at equality).

Write \(g\) for the column of lower virtual counts by color, \(u=M{\bf1}-g\). Thus \(M\ge\max g_c\) and the winding constraints for holes read \[\sum(t_i-\lambda_i)=bQ .\] Let \(P_+,P_-\) be projections onto \({\bf1},Q\), \(a=9/8\), and put \[\nu_i=-2T_i-P_+ r_i/a,\qquad M_0{\bf1}=n{\bf1}+\sum_i\nu_i+g .\] Indeed \(Q^t\nu_i=-p_i\), so the last equality is consistent. \(M_0=n+O(\#\mathrm{objects})\) is real and independent of positions and periods. The table’s charges \(j=(1,1,-1,1,2)\) have sum zero over the configuration, by the winding constraint. At small \(v\), \[G(v)=\frac{j j^t}{2av}+K_0+O(v),\qquad (K_0)_{ij}=\frac{p_i z_j-z_i p_j}{4}-k_{ij} +\nu_i^t f_j-f_i^t\nu_j,\quad G(-v)=-G(v)^t.\] This follows by multiplying the columns using \(A=P_-/(2v)+A_1 v+O(v^3),\ P_+A_1=aP_+\); the bare pole cancels in the minus channel.

A kernel positivity decomposition

Our next task is to bound the residual particle sum uniformly in the spectral period. A positive Fourier form will control high local populations; a separate pole term will control the cumulative integer charge. The following decomposition supplies both controls.

In this calculation only the five species (per color) index the matrices; the symmetric part of \(G(v)\) for \(v>0\) is meant. One can subtract the following pieces:

•\(j j^t\operatorname{sech}^2(Kv)/(2av)\), with \(K\) a sufficiently large constant, all colors;

•on the opposite-color blocks only, the entrywise repulsion \[R(v)=\tanh(v/2)\big(R_1/\cosh v+R_2/\cosh(v/2)\big)+2\tanh v\ e_5 e_5^t ,\] where \[8 R_1=\begin{pmatrix}6&15&0&3&16\\15&22&0&9&26\\0&0&2&0&0\\3&9&0&0&5\\16&26&0&5&5\end{pmatrix},\qquad 8 R_2=2(e_2e_5^t+e_5e_2^t)+5(e_4e_5^t+e_5e_4^t);\]

•on the diagonal, \(\tanh(c v)\), for sufficiently small \(c>0\).

The result divided by \(v\) is an even, smooth, exponentially decreasing positive Fourier form; it even dominates \(c_1\operatorname{sech}(c_2 v)\) times the identity, for some positive constants. The repulsions (second and third pieces divided by \(v\)) have entrywise nonnegative Fourier transforms. Indeed \(\tanh(c v)/v=\int_0^c\operatorname{sech}^2(u v)\,du\), and \(\operatorname{sech}\) has positive Fourier transform (by shifting through one pole), so squares and products here give positive convolutions.

Here are verification details of the matrix inequality. Work in either color channel \(h=\pm1\), i.e. the eigenblock same plus \(h\) times opposite, and first subtract only \(hR(v)\). Call the resulting symmetric matrix \(S_h(v)\). With \(x=e^{v/8}, w=x^8\), put \(D=w^4-w^{-4}\) and \[A_n=w^{-4}(w^2-w+1)(w^3-h)^2.\] Thus the single upper eigenvalue is \(A_n/D\). After removal of the offsets (factor \(x^{8\beta},\ \beta=a_j-a_i\)), the oriented entry of \(G\) in this channel is \[m_{ij}(w)/D - p_i(w^{-1})p_j(w)/(D A_n)+b_{ij}.\] Here \(m_{ij}\) is the divisor exponential sum for bare gas parts, \(p_i\) that from an upper variable plus \(\tau_i D\), and \(b_{ij}\) the hole-direct and crossing entries; explicit arrays are given below. At every \(w^8=1\), \(m_{ij}A_n=p_i(w^{-1})p_j(w)\), since wrapping and the branch choice do not change any powers. At \(w=1, h=1\) the difference vanishes to order at least two because \(p_i(1)=0\) and \(A_n\) has a double zero. Thus, putting \(z=w^{-1}\), \[\begin{split} N&=44\ (h=1),\quad40\ (h=-1),\\ \Delta_+&=(1-z)(1+z)^2(1+z^2)(1+z+z^2)^2(1-z+z^2),\\ \Delta_-&=(1+z)^2(1+z^2)(1-z+z^2)^3 , \end{split}\] the matrix \(4 x^N\Delta_h S_h\) is a Laurent polynomial of parity \(h\) under \(x\mapsto1/x\), with powers at most \(N\) (indeed \(S_h\to1\) at infinity). Denote its coefficients at \(x^k,\ k\ge0\) by \(P_k\), taking only half the constant coefficient. In the Taylor expansion in \(\log x\) the relevant matrices are \[B_p=\sum_{k=0}^N (k/N)^p P_k\] for even \(p\) in the plus and odd \(p\) in the minus channel. \(B_0=128 j j^t\) for the plus channel (here \(j\) is the five-component vector). All the other relevant matrices are positive definite.

We include small algebra details for checking this last sign test. Take \(K_i=(2),(2,3),\varnothing,\varnothing,(2,3)\), respectively, and \(\tau_i=0,0,1,-1,-1\). Let \(d_l=(2,-1,1,-2h,2h,-2h,1,-1)_l\) for residues starting at 0. Put \[L_b(m)= \begin{cases} z^{8-(m\bmod8)}&m\not\equiv0,\\ z^8, 1, (1+z^8)/2&m\equiv0,\quad b>0, b<0, b=0\ \text{respectively}. \end{cases}\] Then \[m'_{ij}=z^4 m_{ij}(w)=\sum_{r\in K_i,s\in K_j,l}d_l L_\beta(l+s-r),\quad u_j=z^4 p_j(w)=\tau_j(1-z^8)+\sum_{s\in K_j,l}d_l L_{\sigma_j}(l+s),\] where \(\sigma_j>0\) on L, negative on D or P. Finally \[b_{ij}=-\tau_i\tau_j\operatorname{sgn}\beta+(1-2h) [\mathbf1_{\tau_i=-1,\,|K_j|=2}\chi(\beta)- \mathbf1_{|K_i|=2,\,\tau_j=-1}\chi(-\beta)]\] with \(\chi\) the positive step (half at zero).

The following table gives integer parts of the five successive leading minors; the last row in each block instead gives integer parts of \(100\sum_{k<N}\sum_j (k/N)^p |P_{k,ij}|\), by rows \(i\). \[\begin{array}{r|rrrrr@{\quad}r|rrrrr} p(+)&1&2&3&4&5&p(-)&1&2&3&4&5\\\hline 2&22&149&542&2294&10192&1&1&11&9&9&3\\ 4&10&32&31&54&18&3&1&9&9&11&9\\ 6&7&15&28&97&174&5&1&12&14&17&21\\ 8&5&13&38&135&211&7&2&15&22&31&59\\ 10&4&12&44&150&177&9&2&17&29&43&122\\ 12&4&13&47&158&144&11&3&18&33&52&185\\ 14&4&13&49&165&125&13&3&17&36&63&238\\ 16&4&14&50&173&122&15&3&17&38&77&289\\ 18&4&14&52&180&133&17&3&16&40&96&347\\ 20&4&14&53&188&156&19&3&16&43&115&412\\ 22&4&15&54&196&188&21&3&16&45&135&482\\ 24&4&15&56&203&227&23&3&16&48&153&550\\ 26&4&15&56&209&271&25&3&15&50&169&614\\ 28&4&15&57&215&318&27&3&15&52&183&672\\ 30&4&15&58&220&366&29&261&374&287&271&231\\ 32&4&15&59&225&414\\ 34&4&15&59&228&461\\ 36&4&15&60&232&506\\ 38&4&15&60&235&549\\ 40&4&15&60&237&589\\ 42&178&295&150&177&382 \end{array}\] Since \(P_N=4I\), each last row proves strict diagonal dominance at its index. The factors \((k/N)^p\) for \(k<N\) decrease with \(p\), so diagonal dominance continues at every larger relevant index. The preceding rows prove positivity of the remaining \(B_p\) by their leading principal minors.

Here is one way of doing the additions/multiplications for the table. Form to order six inclusive \[H_{ij}=\Delta_h(m'_{ij}+b_{ij})-Q_h(z) z^8 u_i(1/z)u_j(z), \quad Q_+=(1+z)^2(1+z^2)\sum_{d=0}^6 z^d,\quad Q_-=1+z^2 .\] For the two channel blocks separately, set \[D_{l,ij}=2[z^{l/8+a_j-a_i}]H_{ij} -2h[z^{l/8}]\Delta_h\left( 2\frac{1-z}{1+z}\left(\frac{z R_1}{1+z^2}+\frac{\sqrt z R_2}{1+z}\right) +2\frac{1-z^2}{1+z^2}e_5 e_5^t\right)_{ij}.\] A missing power means coefficient zero. Starting at \(l=0\) through \(N-1\), use \(D_l+D_l^t\) (so the first is \(4I\)). Indeed expansion of the displayed oriented entry gives \(H_{ij}\), since \((1-z^8)^{-1}\) can be dropped here. To obtain column \(b\) in a minor row, take \[\left\lfloor \frac{\det[\sum_{l=0}^{N-1}(D_{l,ij}+D_{l,ji})(N-l)^p]_{i,j\le b}}{N^{pb}} \right\rfloor ;\] in the last row add \(|D_{l,bj}+D_{l,jb}|(N-l)^p\) for \(l>0\) instead and multiply by \(100/N^p\). These coefficient formulas use only polynomials of small degree.

For some further detail on the common denominators: at an eighth root \(w\) put \(U=\sum_l d_l w^{l-4}=\sum_l d_l w^{4-l}=A_n\). Then \(m_{ij}=U(\sum_{r\in K_i}w^{-r})(\sum_{s\in K_j}w^s)\), and \(p_i(w)=U\sum_{r\in K_i}w^r\) there. Once the \(D\) poles cancel this leaves the factor \((1-z+z^2)(1-hz^3)^2\), with at most a simple pole at \(z=1\) in the plus channel; the repulsion denominators use \((1+z)^2(1+z^2)\). Their combination thus divides \(\Delta_h\). The offsets only give integral powers of \(x\). Symmetrization is odd under \(v\mapsto-v\) before multiplication by \(4x^N\Delta_h\); this gives the stated parity. The plus constant Taylor value is \(2B_0\) and also follows immediately from \(\Delta_+\sim72v\) and the pole. For moments at positive index \(p\), only coefficients strictly above power zero in \(x\) matter, so \((N-1)/8+\max|a_i-a_j|\) controls the truncation in \(z\).

To pass from these coefficient tests to every \(v>0\), expand the Laurent polynomial in \(\log x=v/8>0\). Its parity and the half-constant convention give \[\begin{split} 4x^{44}\Delta_+S_+&=2\sum_{r\ge0} \frac{(44\log x)^{2r}}{(2r)!}B_{2r},\\ 4x^{40}\Delta_-S_-&=2\sum_{r\ge0} \frac{(40\log x)^{2r+1}}{(2r+1)!}B_{2r+1}, \end{split}\] where each line uses its own channel’s \(B_p\). All scalar weights are nonnegative, and \(\Delta_h>0\) for \(x>1\), proving that \(S_h\) is positive definite. Moreover \(B_1\) gives positivity of \(S_-/v\) at zero, while \(B_2\) gives the positive order-\(v\) coefficient of \(S_+\) on the orthogonal complement of its pole; the denominator corrections to \(B_0=128jj^t\) remain in the pole direction. Subtracting the indicated smoothed pole leaves a positive definite order-\(v\) coefficient for sufficiently large \(K\); away from zero this subtraction can be made arbitrarily small by further increasing \(K\). At infinity the result is still \(I+O(e^{-c_3 v})\), \(c_3>0\), so subtracting \(\tanh(c v)\) times identity for sufficiently small \(c\) is also allowed and leaves the asserted lower bound at all frequencies.

Indeed in the plus block the pole removed is \(j j^t\operatorname{sech}^2(Kv)/(a v)\), improving the regular linear coefficient in its pole direction arbitrarily as \(K\) increases. Once this works in a neighborhood of zero, increasing \(K\) only improves the pointwise form there. Similarly one can decrease \(c\) further on compact intervals after ensuring the inequalities near zero and infinity (at infinity first take \(2c<c_3\)).

Estimates and summation of the defect gas

The preceding matrix calculation is only a frequency-by-frequency statement. We must turn it into an absolutely summable particle expansion, continue it to complex periods, and then sum the original count \(M\). We do these operations in that order; the final complex-twist estimate will be obtained only after all three steps.

Take \(R=\Re L\) large, \(|\Im L|\) bounded, \(s\) in a compact subset of the strip \(|\Im s|<\pi\). Constants can depend on these strip bounds. Positions may run along a common contour \(\gamma(r)=r+i\chi(r)\), \(-R/2<r<R/2\), extended by \(\gamma(r+R)=\gamma(r)+L\), with \(\chi\) odd, bounded on the segment, piecewise smooth with Lipschitz constant \(\sigma\) sufficiently small. Split the parametrizing period into boxes of widths between 1 and 2 with populations \(n_d\); put \(m=\sum n_d\). We give details also justifying convergence of the expansions.

Bulk calibration and puncture exponent

The defect sum is bounded after extracting its explicit pressure, determinant and harmonic twist. We now compute these factors and compare them with Proposition 21; this completes Theorem 4.

For the overall determinant, Euler summation gives \[-\sum_{m>0}\log\det A(m\varepsilon)=L/4+O(1).\] Indeed the eigenvalues are \((2\cosh v-1)(\cosh3v\mp1)/\sinh4v\); the negative log determinant extends smoothly at zero, decays exponentially, and its integral on the positive line is \(\pi^2/8\), by integrating \(2[\log(1-e^{-8v})-\log(1-e^{-6v})-\log(1-e^{-v}+e^{-2v})]\). Bounded Euler error follows also on the slightly rotated rays by the derivative bound there.

Put \(I=\int_0^\infty H_*(v)dv\). Poisson summation for the even extension shows \[\varepsilon\left(\tfrac12 H_*(0)+\sum_{m>0}H_*(m\varepsilon)\right) =I+c_* e^{-L}+O(e^{-4\Re L/3})\] for an absolute constant \(c_*\). The transform at positive frequencies \(2\pi k/\varepsilon\) can use the same tilted line shift as above, here passing the first two simple poles at \(i\pi/4,i\pi/3\) (removable poles, if any, simply give zero), which gives this estimate including bounded imaginary parts of the period. For the bulk integral use \(y=e^{-v}\): \[v H_*(v)=\frac{2y^2(1-y^3)^2}{(1-y+y^2)(1-y^8)} =\frac{P(y)}{1-y^{24}}\] with symmetric coefficients \(P_j=P_{24-j}\), sum zero; at \(j<12\) the nonzero ones are \(P_2=P_3=2,\ P_5=P_6=-6,\ P_8=P_9=8,\ P_{10}=2,P_{11}=-6\). Integrate groupwise against \(dv/v\) (absolute convergence by the double zero of \(P\)). Pairing the resulting log products symmetrically about 12 gives \[I=-\sum_{j<12} P_j\log\sin(\pi j/24),\qquad \exp I=8D_0^3/81.\] Here one uses \(\prod_{l\ge0}(l+x)(l+1-x)/(l+1/2)^2=\sin(\pi x)\); e.g. log differentiation gives \(\pi\cot(\pi x)\) by its elementary partial fractions (residue integration on expanding half-integer squares), and the normalization is at \(1/2\). To simplify, the sines at 2 and 10 have product \(1/4\), at 5 and 11 product \((1+\sqrt2)/4\), at 3 and 9 product \(\sqrt2/4\).

Multiply the analytic numerator \(\mathcal N_n(\mathfrak q,s)\) by \(\exp(-c_* n^2\mathfrak q^2)\). On \(|\mathfrak q|=n^{-3/4}\) the bounds proved above apply around the entire circle, with the pressure error after multiplication bounded in log. The correcting factor equals one at zero, so Cauchy’s formula gives \[\mathcal N_n(0,s)=\frac{1}{2\pi i} \int_{|\mathfrak q|=n^{-3/4}} e^{-c_*n^2\mathfrak q^2}\mathcal N_n(\mathfrak q,s) \frac{d\mathfrak q}{\mathfrak q}.\] This bounds the original numerator \(S_{2n}(1)^2X_{n,n}(\kappa=e^s)\). The \(\kappa\)-free part of \(B_n\) times the bulk growth is exactly \(D_0^{-2n}(4D_0^2/9)^{2n^2}\), the squared exponential factor in the denominator bound. The remaining factors besides constants are bounded by \(\left|\exp[L/4+s^2/(4a\varepsilon)]\right|\). Since \(a=9/8\), \(\varepsilon=2\pi\alpha/L\), and \(\alpha=\pi/4\), their powers on this circle satisfy, uniformly on each compact tilt set, \[\left|\exp[L/4+s^2/(4a\varepsilon)]\right| \le C n^{3/8+2\Re(s^2)/(3\pi^2)},\qquad \frac38-2\cdot\frac5{48}=\frac16.\] Thus division by the squared homogeneous Pfaffian gives, locally uniformly in \(|\Im s|<\pi\), \[|E_n(s)|\le C,\quad E_n(s)=X_{n,n}(\kappa=e^s)\,\exp[-(1/6+2s^2/(3\pi^2))\log n].\] At \(s=i\pi/2\) this analytic function is exactly 1. Hence all these puncture polynomials as functions of \(\eta=2\cosh s\) are zero-free in a fixed disk about \(\eta=0\), with Taylor coefficients of each fixed order in \(\log X_{n,n}\) bounded by \(O_k(1+\log n)\) (Cauchy estimates for \(\log E_n\) near the normalization point). Also \[X_{n,n}(1)=n^{1/6+o(1)}\] where 1 denotes \(\kappa=1\), thus fugacity 2. For the lower bound take any subsequence and a further locally uniform limit of \(E_n\) in the strip. It is nonzero at the normalization point and hence nonzero also at points \(s=i t\) with \(t>0\) arbitrarily small. There \(2\cos t<2\), so coefficient positivity gives the lower bound with exponent \(1/6-2t^2/(3\pi^2)\). The upper bound uses \(s=0\) directly.

Here compact subsequential convergence follows e.g. from derivative bounds by Cauchy’s estimates followed by a compact exhaustion and diagonal selection, with holomorphy of the limit. The same bounds at \(i\pi/2\) give a neighborhood where \(|E_n-1|<1/2\) uniformly, and \(\eta\) has a local analytic inverse parameter there since its derivative is nonzero.

Weighted incompatibility from the zero-free disk

Let \(r_0>0\) be a common radius of a zero-free disk for \(F_n(\eta)\) about \(\eta=0\). Available polygons on a fixed balanced cylinder have weights \(w(P)=x_*^{|P|}\). Write \(I(P,Q)\) for incompatibility, including \(I(P,P)=1\).

To use this lemma for oversized polygons, we will force incompatibilities between two such polygons after rotating one of them. If all six rotations of the smaller polygon were compatible with the larger one, their interiors would all lie inside its period-free interior. The next geometric lemma rules this out at radius greater than \(p^{1+\delta}\).

The six-rotation radius bound

Write \(\omega=e^{i\pi/3}\) and \(\Lambda=\mathbb Z[\omega]\) for the triangular face-center lattice.

Removing oversized polygons

The geometric lemma also gives a form that will be used for polygons in a strip. Here a pure lattice period may point in any direction; it need not be the horizontal period of the original balanced cylinder.

In the strip application below, a transverse lattice period of length comparable to the strip height makes each polygon interior period-free. Corollary 25 therefore supplies the same oversized-polygon removal before Proposition 22 is applied to the retained disk system.

Exponential lateral tails in a strip

Before normalizing the finite chord laws, we control excursions that travel sideways while remaining in a thin strip. This estimate also justifies the subsequent lateral exhaustions with a length insertion.

Macroscopic chords in a hexagon

For specificity take regular lattice hexagons of integer side \(R\to\infty\). Say an arc is macroscopic when its diameter is at least \(R/10\). Their absolute critical mass over all boundary sources is \[R^{3/4+o(1)}.\] For the lower bound use sources in the central third of one side. Exits not to the initial bottom side all have macroscopic reach. Flux and comparison of bottom returns with arches in a containing flat strip of \(O(R)\) layers give non-bottom mass at least \(b_{O(R)}\).

For the upper bound per source, impose a cut parallel to its starting side at height \(h=\lfloor R/\log^2R\rfloor\). Flux subtraction bounds the mass of exits removed by this cut by a constant times \(b_h\), since its new exit prefixes are strip bridges. A retained macroscopic path must reach horizontal distance comparable to \(R\), so Lemma 27 bounds its weight by \(Ce^{-c\log^2R}\). Summing over the \(O(R)\) boundary sources proves the upper normalizing exponent.

Under the normalized macroscopic arc weights, bulk vertices (say in the central third) now have visit probability \[R^{-2/3+o(1)}.\] Indeed all visiting paths there qualify, and the surrounding-polygon sums in Proposition 26 lie between disk sums at comparable radii. Proposition 22 gives their exponent. The same upper bound holds throughout the hexagon. Summing the probabilities yields the mean visited mass, or length, \[\mathbb E |\gamma|=R^{4/3+o(1)} .\]

A fixed-source strip version

There is also an annealed dimension statement with one actual boundary point fixed. In the full planar strip of \(h\) layers take all the critical exits from a fixed bottom port (thus bridges to the top or arches back to bottom), conditioned to get at least one quarter of the way up. Then \[\mathbb E |\gamma|=h^{4/3+o(1)} .\] Changing this to any fixed positive fraction strictly below one half gives the same result. The exit mass of this ensemble before normalization is \(h^{-1/4+o(1)}\): lower by the bridges, upper for the arches by the height-tail bound. This conditioning could also confine horizontal range to \(h\log^2 h\) about the start without changing the exponent. For bridges alone the argument gives the mean upper bound \(h^{4/3+o(1)}\) (and hence the usual in-probability upper-exponent bound), not by itself the matching bridge lower bound.

The pointwise estimate behind this statement will also be needed for the bridge lower bound. It sums over all boundary positions, rather than fixing a source.

We finish Proposition 3 by averaging over horizontal sources. In a finite truncation \(D\), counting every visited vertex gives the exact identity \[\sum_{v\in D} A_D(v) =\sum_{\gamma\text{ an ordered excursion in }D} |\gamma|x_*^{|\gamma|}.\] Divide by horizontal width and let that width tend to infinity at fixed \(h\). Lemma 28 bounds the result above by \(h^{13/12+o(1)}\). A central band of \(\asymp h\) layers supplies the same lower exponent. For the ensemble conditioned to reach \(\vartheta h\) layers above its starting wall, where \(0<\vartheta<1/2\) is fixed, choose this band symmetric about the midline and more than \(\vartheta h\) layers from either wall. Every path contributing a visit to that band then meets the conditioning event, from either horizontal starting side.

The slanted side sources disappear in this average: there are only \(O(h)\) of them, and the tail estimate in the preceding proof bounds each one’s length mass at fixed \(h\). Their contribution divided by width therefore vanishes.

Each horizontal wall has one source port per unit length. Translation invariance and top–bottom symmetry therefore identify the limiting average with twice the fixed-source length weight. One can first restrict to bounded path range, pass to the large-width limit, and then remove that restriction using the exponential tails. Thus the conditioned ensemble has unnormalized length weight \(h^{13/12+o(1)}\). Its mass is \(h^{-1/4+o(1)}\) by Proposition 18, giving the asserted mean. The same tails make the discarded first moment at horizontal range \(h\log^2 h\) smaller than every fixed power of \(h\), proving the optional confinement statement.

Deletion and first contact

An active outer path is phased by \(\exp(i\sigma W)\) as before. It may be stopped just before first attempting entry to \(v\), without using a weight there; call such prefixes \(\eta\). The phased full exits through \(v\) sum to the sum on these prefixes by deletion flux. Likewise for first hit (meaning first attempted vertex, not taking its weight) of another obstacle. For disjoint obstacles \(v,X\), the phased sum of full exits hitting both equals the sum of prefixes to first \(X\) that have visited \(v\), plus prefixes to first \(v\) that have visited \(X\): just use deletion for \(v,X,v\cup X\). All actual path intersections below mean shared visited vertices.

At fixed \(h\) these operations in the unbounded strip can be done by absolute convergence. Indeed embed the strip by projection into a tilted cylinder, with a period along another tiling direction and large enough vertical shift for injectivity on the strip. On that cylinder strip-bridge weights in the longitudinal direction decay exponentially in the number of layers, as above. All SAWs from a fixed vertex there have absolutely summable critical weights with exponential tails in longitudinal span \(D\): lengths are \(O_h(D+1)\), and the weak-bridge unfolding count (decomposition at extremal heights on the two halves, as above) uses subexponentially many lists with sublinear number of spans, whose sum is at least \(D\). Positive weak bridges again extend by boundedly many vertices to cut lines without repetitions and each costs at most a constant times exponential decay in its span. Thus the loss besides that exponential is subexponential. This gives absolute convergence also with polynomial factors and for connected finite products of paths with one site anchored. Flux calculations can equivalently truncate laterally first (also when a fixed finite crosscut is removed).

Let \(J\) be the absolute sum for a bridge and an arch (arch counted once, on either side) intersecting each other with at least one through \(v\), and \(J_{CC}\) the analogous ordered sum for two bridges. Each pair receives the product of its two path weights, so a vertex shared by the two independently chosen paths occurs in both factors. These sums are finite by the anchored convergence just proved. Our first target is \[ J\le C(J_{CC}+m). \tag{15}\] Once this comparison is established, lateral tails will bound \(J_{CC}\) in terms of \(m\), while translations will bound \(J\) below in terms of \(u\). The following complex sums are introduced solely to prove (15).

The marked tripod identity

Use triples of boundary ports \(a,b,c'\), where \(b<c'\) are ordered on one horizontal side and \(a\) is on the opposite. Positive order is east along the bottom, reversed along the top. We sum over both choices of side. A tripod consists of simple arms from these ports to a center vertex, mutually disjoint until the center, all weights included there once, the arm turn \(W_j\) from \(j\) measured up to arrival (before turning at the center). By planarity their cyclic order is forced. Since joining \(b\) to \(a\) or \(c'\) to \(a\) forms a bridge (turn zero in total), and the join turns are \(+\pi/3,-\pi/3\) respectively, \[e^{i\sigma W_a}=p\,e^{i\sigma W_b}=\bar p\,e^{i\sigma W_{c'}}.\] Figure 4 shows the first-contact construction. Only the cyclic order, rather than the pictured shape, enters the phase relations.

Let \(L_*\) sum all tripods containing \(v\), using the first (common) phase \(e^{i\sigma W_a}\).

The following sums use independent spectator and active paths, with only the active path phased. A full exit includes all its visited vertex weights; a prefix \(\eta\) stops just before \(v\) and omits its weight. The port order and the sum over both choices of side are as above.

Set \(Z=Z_b+Z_{c'}\). In \(E_C\) and \(R_C\), the active source may lie on either boundary and may coincide with a spectator endpoint.

Hitting a marked spectator is represented by first contact by deletion. This gives exactly the contact tripod weights; in particular \(A,p Z_b,\bar p Z_{c'}\) in common phase restrict \(L_*\) to \(v\) not exclusively on the open arm \(a,b,c'\), respectively. The contact representation uses the free third edge for the incoming active prefix, since its source differs from the spectator endpoints. When the spectator avoids \(v\), the term in which \(v\) is visited before first contact is instead the complementary restriction in \(L_*\), with phase depending on the active arm. Thus \[R_A=E_A+A-L_*,\qquad R_C=E_C+Z-2k L_*.\] This uses the both-obstacles formula above; if the active source coincides with a bridge spectator endpoint, all exits and prefixes to \(v\) hit that spectator with no possible \(v\) before first contact, consistent with the second equation.

For another balance, let the spectator now be a prefix \(\eta\) from the top weighted with conjugate phase, and grow the opposite active path in \(S\setminus\{v\}\). Its valid ends are an original outer boundary port or an attempt at \(v\), stopped before taking its weight. Revisit pairs from an outer source still cancel, with or without deleting the spectator vertices. Use also the bottom spectator and top active balance, add, and take real parts. The real sum of products for valid leaves hitting those spectators is \[\Re(2c R_A+R_C)+2P,\qquad P=\Re\!\!\sum_{\eta_B\cap\eta_T\ne\varnothing} w(\eta_B) w(\eta_T) e^{i\sigma(W_B-W_T)}\] where subscripts denote the lower/upper source and \(w\) denotes absolute path weight. Outer leaves are arches avoiding \(v\) (both orientations, averaging to phase \(c\)) or bridges avoiding \(v\) (phase \(1\)); the other leaves are the intersecting prefixes.

Write \(u_B,u_T\) for the once-counted arch weights through \(v\) on the bottom and top, so \(u=u_B+u_T\). By deletion, the two full prefix sums are the nonnegative real numbers \[M_B=m+2cu_B,\qquad M_T=m+2cu_T.\] Disjoint opposite-side prefixes join at \(v\) to give each bridge through \(v\) once. Their product omits exactly the weight \(x_*\) at \(v\), and their arrival-turn difference is \(\pm\pi/3\), with real phase \(k\). Subtracting these disjoint pairs from the full product gives \[ P=M_BM_T-\frac{k}{x_*}m\ge-\frac{k}{x_*}m. \tag{16}\]

At first contact the two shoulders from opposite sides form a bridge \(\lambda\) avoiding \(v\). The incoming active edge is free: neither the edge toward the spectator’s source nor the edge toward its stopped-at \(v\) could have brought the opposite active path to its first contact. The spectator suffix starts along the third edge from the center and goes to \(v\), avoiding \(\lambda\). The bridge takes the center weight once; the suffix takes neither that weight nor the weight at \(v\). Conversely these objects give all contacts once per choice of active side.

Write \(\delta=\pm\pi/3\) for the bottom-to-top join turn at the center, and \(W'\) for the suffix turn starting along its outgoing edge. The shoulder-turn difference is \(-\delta\); the turn into the suffix from the top is \(\delta\), and from the bottom \(-\delta\). The two phase factors therefore add to \(2g e^{-i\sigma W'}\). For fixed \(\lambda\), the real suffix sum without \(2g\) is exactly the real sum of prefixes to \(v\) in \(S-\lambda\) from the new boundary ports on the unused outward edges of \(\lambda\). Length zero is included when appropriate.

Write \(H(S;v)\) and \(H(S-\lambda;v)\) for the real prefix sums to \(v\) from the respective whole boundaries. Original prefixes avoiding \(\lambda\) are unchanged, so the new-boundary suffix sum equals the original-boundary prefix sum hitting \(\lambda\), minus \[\Delta_\lambda=H(S;v)-H(S-\lambda;v)\ge0.\] For the inequality, reverse to exits from \(v\) and use the positive nesting expansion of Proposition 26, whose final multipliers are \(3+(2/x_*)\sum_Q c_{m_Q}w(Q)\). The domain \(S-\lambda\) is filled: a removed bridge connected to the outer boundary cannot enter the strict interior of a polygon avoiding it. For fixed \(\lambda\), take the nesting comparison first in convex lateral truncations with and without its vertices, and then pass to the strip; side contributions anchored at \(v\) vanish by the convergence already proved. When summing \(w(\lambda)\Delta_\lambda\), use its representation as hit prefixes minus new-boundary prefixes. Both are connected terms; no sum of disconnected boundary terms is needed. The first-contact calculation consequently gives \[\Re(2cR_A+R_C)+2P=2g\left[\Re R_C-\sum_{\lambda\not\ni v}w(\lambda)\Delta_\lambda\right].\] Using the earlier equations (the \(L_*\) terms cancel since \(\phi k=c\)), \[2c\Re(A+E_A)+2P\le \phi\Re(E_C+Z).\]

Comparison of intersecting path pairs

We now compare the phased expressions with the positive pair sums. Including only bridge exits as active paths gives \(\Re(A+E_A)\ge J\); the remaining exits have positive real weights. Also \(\Re(E_C+Z)\le2cJ+2J_{CC}\): allow active sources coinciding with a marked spectator endpoint, adding nonnegative real weight, and combine both orientations of every active path. Together with (16), the last phased inequality becomes \[2c(1-\phi)J\le2\phi J_{CC}+\frac{2k}{x_*}m.\] Since \(\phi<1\), this proves (15).

Let \(\mu_h\) be the unnormalized measure on bottom-to-top bridge shapes from one fixed bottom port, of total mass \(b_h\). Write \(X(\beta)\) for the largest horizontal distance of a visited vertex from that source, as in Lemma 27, and set \(T=h\log^2h\) for large \(h\). If both bridge shapes have \(X\le T\), then for any fixed marked bridge the other shape has at most \(CT\) translations hitting it. Choosing which member contains \(v\) only changes the constant. Thus \[J_{CC}\le CTb_hm+R_h,\] where \(R_h\) is the contribution of pairs with at least one \(X>T\).

We bound this remainder absolutely, without an estimate on \(m\). A fixed-source shape \(\beta_1\) has at most \(|\beta_1|\) translations placing a visited vertex at \(v\), and \(|\beta_1|\le C(h+1)(X_1+1)\), where \(X_i=X(\beta_i)\). For each such translate, a second shape has at most \(C(X_1+X_2+1)\) translations hitting it. Hence, uniformly in \(v\), \[\begin{align*} R_h&\le C(h+1) \iint_{\max(X_1,X_2)>T}(1+X_1+X_2)^2 \,d\mu_h(\beta_1)\,d\mu_h(\beta_2)\\ &\le C(h+1)(T+h+1)^2e^{-cT/(h+1)}=O(h^{-100}). \end{align*}\] The second line follows by integrating the exponential tail in Lemma 27 for the first two moments of \(X\), using \(b_h\le1\) and splitting according to which \(X_i>T\). Since \(b_h=h^{-1/4+o(1)}\), we obtain \[ J_{CC}\le h^{3/4+o(1)}m+O(h^{-100}). \tag{17}\]

Translation hits on a deep arch

The upper pair bound pays one bridge crossing mass for each allowed horizontal translation. We need a matching lower bound for bridges hitting each arch through the marked vertex. Here the error must be uniform in the arch, whose length and horizontal span are unrestricted.

The bridge first moment

Lemma 29 gives \(J\ge h^{3/4-o(1)}u\). Combining this with (15) and (17) yields \[u\le h^{o(1)}m+O\big(h^{-100-3/4+o(1)}\big).\] On the other hand, \(2(u+m)=A_{S_h}(v)\): the ordered excursion convention counts each path twice. Lemma 28 therefore gives \(u+m=h^{1/12+o(1)}\) in the central third, uniformly in \(v\). The negligible additive error above cannot carry this mass, so \[\begin{align*} m(v)&=h^{1/12+o(1)} &&\text{uniformly in the central third},\\ m(v)&\le h^{1/12+o(1)} &&\text{throughout the strip}. \end{align*}\]

To pass from a marked vertex to a fixed source, let \(\mathcal V_h\) contain one representative of each horizontal-translation orbit of strip vertices. Translating a bridge’s bottom source to the fixed source identifies \(m(v)\) with the weighted number of visits to the orbit of \(v\). Thus the unnormalized fixed-source length mass is exactly \[\ell_h=\sum_{v\in\mathcal V_h}m(v).\] There are \(O(h)\) such orbits, with order \(h\) represented in the central third. The pointwise upper and lower estimates give \(\ell_h=h^{13/12+o(1)}\). Finally, dividing by \(b_h=h^{-1/4+o(1)}\) gives \[\mathbb E|\gamma|=\frac{\ell_h}{b_h}=h^{4/3+o(1)}.\] Lemma 27 makes both the discarded mass and discarded first moment at horizontal range \(h\log^2h\) negligible, so the same conclusion holds with that confinement. This completes Theorem 1.