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

The ferromagnetic symmetric-group ($S_Q$) permutation model is the replica statistical model whose ordering transition encodes the entanglement transition of random tensor networks. A recent star-triangle (finite-basis) duality analysis by Ohzeki (arXiv:2607.14917; published as Prog. Theor. Exp. Phys. 2026, ptag162) estimates its critical bond dimension on the honeycomb lattice at $D_c^{hex}=2.634929344884\cdots$ and, via a single-bond mutual-duality relation, the paired triangular estimate $D_c^{tri}=1.475661534848\cdots$. Independently, the direct triangular block condition reduces in the replica limit $Q\to 0$ to $\gamma+\mathcal{S}(D)=0$ with $\mathcal{S}(D)=\sum_{r\geq 1}b_r D^{-2r}$ a factorially divergent asymptotic series whose Farahat-Higman coefficients $b_r$ are known exactly through $r=24$; Borel-Pade resummation gives only a diagnostic root $1.48463(10)$, which the paper explicitly declines to adopt as a critical-point estimate. The open problem is the full resummation of this series: a closed form or exact representation of $\mathcal{S}(D)$, or a rigorous summability result with certified error, yielding the definitive root of the direct condition and deciding whether the direct projection independently confirms the mutual-dual paired estimate or whether the roughly 0.6% offset of the current diagnostic reflects a genuine systematic discrepancy between the two finite-basis duality projections. No resolution or adjacent result addressing this question is known; the problem is open.

Background

Replica analyses of random tensor networks and of monitored random quantum circuits map disorder-averaged entanglement moments onto classical statistical models whose elementary degrees of freedom are permutations. In the resulting permutation model, each site $i$ of a graph $G=(V,E)$ carries a variable $\sigma_i$ in the symmetric group $S_Q$ (the group of all permutations of $Q$ objects, where $Q$ is the replica index), and the partition function is $Z_G(D,Q)=\sum_{\{\sigma_i\}}\prod_{\langle ij\rangle}f_D(\sigma_i\sigma_j^{-1})$ with $f_D(g)=D^{C(g)}$, where $C(g)$ is the number of cycles of the permutation $g$. The parameter $D$ plays a double role: it is a ferromagnetic coupling (one writes $D=e^K$, so larger $D$ means lower effective temperature) and, in the quantum-information application, the bond dimension, i.e. the Hilbert-space dimension carried by each bond of the underlying random tensor network. The replica limit analytically continues $Q$ from positive integers, and the physical quantity is the coefficient linear in $Q$ at $Q=0$, exactly as in replica treatments of spin glasses. The model's ordering transition separates a disordered phase, corresponding to area-law entanglement of the network, from an ordered phase, corresponding to volume-law entanglement, which is further interpreted as a quantum-error-protecting phase. At $Q=2$ the model reduces to the ferromagnetic Ising model with $D=e^{2K_{\rm Ising}}$, a useful normalization check: the $Q=2$ critical bond dimensions reproduce the exact triangular and honeycomb Ising values $\sqrt{3}$ and $2+\sqrt{3}$.

Duality is the classical analytic tool for locating two-dimensional critical points: a Fourier-transform relation maps a planar lattice model to a model on the dual lattice, and in self-dual cases the equality of the original and dual principal Boltzmann factors (the weight of the configuration in which neighboring variables agree) identifies the critical point, the Kramers-Wannier argument for the Ising model. For random and replicated systems the principal-factor condition is only a scalar projection of the exact duality relation: predictive, but not exact. The established refinement, developed for random-bond Ising systems on the triangular-hexagonal pair by Nishimori and Ohzeki and extended to regular lattices by Ohzeki, is to impose the condition on a finite basis larger than a bare bond; for the triangular/honeycomb pair the appropriate block is the star-triangle block (a triangle of bonds on one lattice maps to a three-legged star on the dual lattice, the local move underlying the exact triangular-honeycomb Ising duality). For the permutation model, the single-bond replica condition $\log D=\psi(D)+\gamma/2$, with $\psi$ the digamma function (the logarithmic derivative of the Gamma function) and $\gamma$ the Euler-Mascheroni constant, gives the bare estimate $D_0=1.882008326950\cdots$ for self-dual lattices; exact real-space renormalization on self-dual hierarchical lattices (self-similar fractal lattices on which the renormalization transformation closes exactly) with extrapolation toward the replica limit supports that estimate.

The source paper extends the finite-basis analysis to the non-self-dual triangular/honeycomb pair. On the honeycomb side, the star-triangle (Y-block) principal-factor condition collapses in the replica limit to the closed equation $\psi(D^3)=3\psi(D)+\gamma$, whose positive solution is $D_c^{\rm hex}=2.634929344884\cdots$; the single-bond mutual-duality relation between the two mutually dual lattices, $\log D_{\rm tri}+\log D_{\rm hex}=\psi(D_{\rm tri})+\psi(D_{\rm hex})+\gamma$, then yields the paired triangular estimate $D_{\rm tri}^{\rm pair}=1.475661534848\cdots$. The direct triangular face condition is subtler. Its dual principal factor is the constrained three-permutation sum

$$\mathcal{Y}_Q(Q,D)=\frac{1}{Q!}\sum_{g_1g_2g_3=e}D^{C(g_1)+C(g_2)+C(g_3)}=\sum_{\lambda\vdash Q}\frac{(D)_\lambda^3}{h_\lambda^2},$$

where the first sum runs over triples $g_1,g_2,g_3\in S_Q$ with $g_1g_2g_3=e$, and the second is a sum over Young diagrams $\lambda$ of $Q$ boxes, with $(D)_\lambda=\prod_{\square\in\lambda}(D+c(\square))$ the content product ($c(\square)$ is the column index minus the row index of the box) and $h_\lambda$ the product of hook lengths. Factoring out the leading behavior gives $\mathcal{Y}_Q=\frac{D^{3Q}}{\Gamma(Q+1)}\left[1+\sum_{r\geq 1}c_r(Q)D^{-2r}\right]$, and the classical Farahat-Higman stability theorem (the structure constants of the center of the $S_Q$ group algebra are polynomials in $Q$) implies that each $c_r(Q)$ is a polynomial in $Q$. Writing $b_r=c_r'(0)$, the replica limit of the direct triangular block condition $D^{3Q}=\mathcal{Y}_Q(Q,D)$ becomes

$$\gamma+\mathcal{S}(D)=0,\qquad\mathcal{S}(D)=\sum_{r\geq 1}\frac{b_r}{D^{2r}},$$

with $b_1=-3/2$, $b_2=5/4$, $b_3=-41/6$, $b_4=2751/40$, $b_5=-2057/2$, and the coefficients known exactly through $r=24$. The series is factorially divergent at the physical value of $D$. Its Borel and Borel-Leroy resummations (the Borel transform divides the coefficients by $\Gamma(r+\beta+1)$, is extrapolated by Pade rational approximants, and is integrated against $e^{-t}$; approximants with poles on the positive Borel axis are discarded) cluster near $D\approx 1.4846$, giving the diagnostic $D_{\rm tri}^{\rm direct}=1.48463(10)$. The paper explicitly declines to use this diagnostic as the critical-point estimate, because a Pade diagnostic does not settle the value; the full resummation of the series is left open.

Problem Statement

Determine the full resummation of the direct triangular block condition for the $Q\to 0$ symmetric-group permutation model, and thereby the definitive direct estimate of the triangular-lattice critical bond dimension. Concretely, for the asymptotic series $\mathcal{S}(D)=\sum_{r\geq 1}b_rD^{-2r}$, where $b_r=c_r'(0)$ are the replica-limit coefficients of the constrained three-permutation sum $\mathcal{Y}_Q(Q,D)=\sum_{\lambda\vdash Q}(D)_\lambda^3/h_\lambda^2$ (exactly known through $r=24$): obtain a closed form or exact representation for $\mathcal{S}(D)$, equivalently for $\left.\partial_Q\log\mathcal{Y}_Q(Q,D)\right|_{Q=0}$, or failing that a resummation with a certified, rigorous error bound, sufficient to determine the root of $\gamma+\mathcal{S}(D)=0$ to definitive accuracy (here a root pinned to within $10^{-3}$, an order of magnitude below the roughly $0.6\%\approx9\times10^{-3}$ offset at issue); a determination by resummation must resolve the status of the positive-axis poles seen in Pade approximants of the Borel transform, establishing whether $\mathcal{S}(D)$ is Borel (or Borel-Leroy) summable at the relevant $D\approx 1.48$. The answer must state whether this direct estimate confirms the mutual-dual paired estimate $D_{\rm tri}^{\rm pair}=1.475661534848\cdots$, or whether the roughly $0.6\%$ offset already visible in the Borel-Pade diagnostic $D_{\rm tri}^{\rm direct}=1.48463(10)$, about ninety times the quoted Pade scatter, reflects a genuine systematic discrepancy between the two finite-basis duality projections: the direct estimate confirms the paired estimate if and only if the definitive direct root agrees with $D_{\rm tri}^{\rm pair}$ within $10^{-3}$, and otherwise the offset is a genuine systematic discrepancy. Completion of this problem is the definitive direct determination together with the consistency verdict; the source's broader suggested extensions (an explicit duality analysis of the statistical-mechanical models of random quantum circuits on square, triangular, and honeycomb lattices; other non-self-dual lattices and higher replica numbers; the connection to quantum-error-correction thresholds) are open research directions without sharp completion boundaries, and are outside the scope of this problem.

References

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