Full resummation of the direct triangular block condition for the symmetric-group permutation model

Ohzeki's star-triangle duality analysis of the symmetric-group permutation model (the statistical-mechanical model whose ordering transition encodes the entanglement transition of random tensor networks) estimates the critical bond dimension on the honeycomb lattice, $D_{\rm c}^{\rm hex}=2.634929344884$, from a closed digamma-function condition, and on the triangular lattice, $D_{\rm c}^{\rm tri}=1.475661534848$, from the associated mutual-dual single-bond relation. The independent direct triangular block condition, however, reduces in the replica limit to $\gamma+\mathcal{S}(D)=0$, where $\gamma$ is the Euler-Mascheroni constant and $\mathcal{S}(D)=\sum_{r\geq 1}b_{r}/D^{2r}$ is a factorially divergent asymptotic series whose coefficients come from the Farahat-Higman stable expansion of a constrained three-permutation Young-diagram sum; Borel-Pade resummation yields only a diagnostic $D_{\rm tri}^{\rm direct}=1.48463(10)$, which the source explicitly declines to use as a critical-point estimate. The open problem is the definitive evaluation of this condition: an exact or certified evaluation of the replica-limit block coefficient $\mathcal{S}(D)$ at the physical bond dimension, the resulting definitive root of $\gamma+\mathcal{S}(D)=0$, and the uniqueness status of the resummation there. The resummation of this condition is unaddressed in the cited literature.

Background

The symmetric-group permutation model is a classical lattice spin model in which the variable at each site is a permutation $g\in S_{Q}$ (the symmetric group on $Q$ letters, where $Q$ plays the role of the replica index) and neighbouring sites interact through the edge weight $f_{D}(g)=D^{C(g)}$, where $C(g)$ is the number of cycles of $g$ and $D$ is a ferromagnetic coupling. In replica analyses of random tensor networks and monitored random quantum circuits, entanglement entropies are mapped to exactly such permutation models: the bond dimension $D$ of the network becomes the coupling, and the ordering transition of the permutation model (ferromagnetic versus paramagnetic) reproduces the entanglement transition between volume-law and area-law phases of the quantum system (phases in which the entanglement entropy of a spatial region grows proportionally to the region's volume, or only to its boundary area, respectively) [Vasseur et al. 2019; Skinner et al. 2019]. The critical bond dimension at which this transition occurs is therefore of direct quantum-information interest: below it the associated random network or circuit cannot sustain long-range entanglement, and the volume-law phase is identified with a quantum-error-correcting regime.

In two dimensions the permutation model has no exact solution for general $Q$ (only $Q=2$ reduces to the Ising model), so critical points are estimated analytically by duality. The Kramers-Wannier argument relates the high-temperature expansion of a planar lattice model to the low-temperature expansion of the model on the dual lattice, and equating the principal Boltzmann factors (the weight of the maximally ordered local configuration and its dual image) gives a scalar estimate of the critical point. For non-self-dual or replicated (quenched-disordered) systems this principal-factor condition is not exact but has proved highly predictive, and for the triangular-honeycomb pair of mutually dual lattices the natural finite-basis upgrade replaces the single bond by the star-triangle block, the local Y-shaped three-bond cluster whose internal summation relates triangular- and honeycomb-lattice models; this strategy was established for random-bond spin glasses [Nishimori and Ohzeki 2006].

Ohzeki applied this program to the permutation model in two steps. A first single-bond duality analysis directly in the symmetric group [Ohzeki 2023] gave the bare estimate $\log D=\psi(D)+\gamma/2$ with root $D_{0}=1.882008326950\ldots$, where $\psi$ is the digamma function and $\gamma$ the Euler-Mascheroni constant; a numerical real-space renormalization study on hierarchical lattices [Ito, Matsuo and Ohzeki 2026] supported the duality prediction after extrapolation toward the replica limit. The star-triangle refinement of [Ohzeki 2026] works in the replica limit, meaning the coefficient linear in $Q$ at $Q=0$, extracted by analytic continuation of identities that hold at positive integer $Q$. The honeycomb Y-block condition closes exactly in standard functions, $\psi(D^{3})=3\psi(D)+\gamma$, with root $D_{\rm c}^{\rm hex}=2.634929344884\ldots$, and the mutual-dual single-bond relation between the two lattices then yields the triangular estimate $D_{\rm c}^{\rm tri}=1.475661534848\ldots$.

The direct triangular block condition, however, does not close. The constrained three-permutation sum around a triangular face evaluates, by the non-Abelian Fourier transform of the symmetric group (the expansion of central functions in the irreducible-character basis of $S_{Q}$), to the Young-diagram sum $\mathcal{Y}_{Q}(Q,D)=\sum_{\lambda\vdash Q}(D)_{\lambda}^{3}/h_{\lambda}^{2}$, where $\lambda$ runs over partitions of $Q$ (labeling irreducible representations of $S_{Q}$), $(D)_{\lambda}=\prod_{\Box\in\lambda}(D+c(\Box))$ is the content product over boxes of the Young diagram ($c(\Box)=j-i$ for a box in row $i$, column $j$), and $h_{\lambda}$ is the hook-length product, with representation dimension $d_{\lambda}=Q!/h_{\lambda}$. Extracting the replica limit uses Farahat-Higman stability, the theorem that structure coefficients of the center of the symmetric-group ring are polynomial in $Q$ at fixed degree [Farahat and Higman 1959]: after factoring out $D^{3Q}/\Gamma(Q+1)$, each coefficient $c_{r}(Q)$ is a polynomial in $Q$, and the replica-limit condition becomes $\gamma+\mathcal{S}(D)=0$ with $\mathcal{S}(D)=\sum_{r\geq 1}b_{r}/D^{2r}$ and $b_{r}=c_{r}'(0)$. This is an exact algebraic condition on the replica limit, but its series is asymptotic at the physical root $D\approx 1.48$: the first coefficients are $b_{1}=-3/2$, $b_{2}=5/4$, $b_{3}=-41/6$, $b_{4}=2751/40$, $b_{5}=-2057/2$, growing factorially, with the list computed through $r=24$. Borel-Pade resummation (replacing the Borel transform of the series by a Pade rational approximant before the inverse Borel integral, and discarding approximants with poles on the positive Borel axis) gives a stable diagnostic root $D_{\rm tri}^{\rm direct}=1.48463(10)$, which the author explicitly declines to use as the critical-point estimate. What is missing is a definitive evaluation of the condition.

Problem Statement

For the ferromagnetic symmetric-group permutation model with edge weight $f_{D}(g)=D^{C(g)}$ on the triangular lattice, give a definitive evaluation of the direct triangular block condition in the replica limit.

Specifically, evaluate the replica-limit triangular block coefficient $\mathcal{S}(D)=\left.\partial_{Q}\log\mathcal{Y}_{Q}(Q,D)\right|_{Q=0}-3\log D-\gamma,\qquad \mathcal{Y}_{Q}(Q,D)=\sum_{\lambda\vdash Q}\frac{(D)_{\lambda}^{3}}{h_{\lambda}^{2}},$ (the coefficient linear in $Q$ at $Q=0$; its large-$D$ asymptotic expansion is $\mathcal{S}(D)\sim\sum_{r\geq 1}b_{r}/D^{2r}$ with $b_{1}=-3/2$, $b_{2}=5/4$, $b_{3}=-41/6$, $b_{4}=2751/40$, $b_{5}=-2057/2,\ldots$) either in closed form or with certified convergent control at the physical bond dimension $D\approx 1.5$, and thereby determine the definitive root of the exact condition $\gamma+\mathcal{S}(D)=0,$ replacing the Borel-Pade diagnostic $D_{\rm tri}^{\rm direct}=1.48463(10)$ with an exact or rigorously certified value, subject to the outcome of the uniqueness characterization required below.

The answer must additionally characterize the uniqueness of the resummation at the physical root: establish Borel summability of the series there (so that the resummed value is unambiguous), or demonstrate a genuine ambiguity (for example, non-Borel summability or positive-axis singularities of the Borel transform) and state what the ambiguity implies for the definiteness of the direct triangular condition.

The single objective uniting these deliverables is to put the direct triangular block condition on the same footing as the honeycomb block condition $\psi(D^{3})=3\psi(D)+\gamma$, which closes exactly in standard functions, with a numerically determined replica-limit root $D_{\rm c}^{\rm hex}=2.634929344884\ldots$: to convert the direct condition from a numerically diagnosed relation into a definitively evaluated one. As in all principal-factor duality arguments, the result is the definitive value of this duality estimate, not an exact critical point of the underlying model.

References

1. M. Ohzeki, Star-triangle duality estimates for triangular and honeycomb permutation models, Prog. Theor. Exp. Phys. 2026, ptag162 (2026), doi:10.1093/ptep/ptag162, arXiv:2607.14917, https://arxiv.org/abs/2607.14917
2. M. Ohzeki, Duality analysis in symmetric group and its application to random tensor network model, arXiv:2310.14140 (2023; revised 2024), https://arxiv.org/abs/2310.14140
3. R. Ito, T. Matsuo, M. Ohzeki, Analysis of critical points in a permutation model on hierarchical lattices by real-space renormalization group, arXiv:2605.25683 (2026), https://arxiv.org/abs/2605.25683
4. M. Ohzeki, Cavity method for permutation models on Cayley trees, arXiv:2606.17751 (2026), https://arxiv.org/abs/2606.17751
5. M. Ohzeki, Finite-basis duality estimate for the surface-code threshold under correlated bit-flip errors, arXiv:2606.04723 (2026), https://arxiv.org/abs/2606.04723
6. H. K. Farahat, G. Higman, The centres of symmetric group rings, Proc. R. Soc. Lond. A 250, 212-221 (1959), doi:10.1098/rspa.1959.0060, https://doi.org/10.1098/rspa.1959.0060
7. H. Nishimori, M. Ohzeki, Location of the multicritical point for the Ising spin glass on the triangular and hexagonal lattices, J. Phys. Soc. Jpn. 75, 034004 (2006), doi:10.1143/JPSJ.75.034004, https://doi.org/10.1143/JPSJ.75.034004
8. R. Vasseur, A. C. Potter, Y.-Z. You, A. W. W. Ludwig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B 100, 134203 (2019), doi:10.1103/PhysRevB.100.134203, https://doi.org/10.1103/PhysRevB.100.134203
9. B. Skinner, J. Ruhman, A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019), doi:10.1103/PhysRevX.9.031009, https://doi.org/10.1103/PhysRevX.9.031009