Jordan-cell structure of the critical site-percolation logarithmic CFT and its universality with bond percolation

The dilute $A_2^{(2)}$ loop model at crossing parameter $\lambda=\pi/3$ is an exactly solved regularization of critical site percolation on the triangular lattice, the dilute logarithmic minimal model $\mathcal{DLM}(2,3)$ with central charge $c=0$. In 2019, Morin-Duchesne and Pearce derived its fusion hierarchy, $T$-system and $Y$-system and observed that solving the associated Thermodynamic Bethe Ansatz equations would test whether site percolation and critical bond percolation on the square lattice (the dense $A_1^{(1)}$ loop model $\mathcal{LM}(2,3)$) lie in the same universality class as logarithmic conformal field theories — a sharper question than comparing spectra, because two logarithmic theories can share every conformal weight yet differ in the Jordan cells of $L_0$ inside their reducible indecomposable Virasoro modules. By 2023–2025 the spectra question was settled by the authors themselves: the conformal weights and the cylinder and torus conformal partition functions of the two models coincide exactly. What remains open, explicitly identified as the next step in the 2023 solution and untouched since, is the Jordan-cell structure itself. The problem is to determine, in the scaling limit of the dilute $A_2^{(2)}$ percolation model in each standard module, which conformal weights are coupled in $L_0$ Jordan cells and with which logarithmic couplings, and to decide whether this structure coincides with the Jordan-cell structure known for bond percolation — completing, or refuting, the strong universality claim for the percolation logarithmic CFT.

Background

Two-dimensional percolation asks for the geometry of the clusters formed when each site (site percolation) or bond (bond percolation) of a lattice is independently occupied with probability $p$. At the critical point, bond percolation on the square lattice and site percolation on the triangular lattice are conformally invariant: crossing probabilities are given by Cardy's formula and cluster boundaries converge to Schramm–Loewner evolution paths with parameter $\kappa=6$ (the traces of planar Brownian motion), so the underlying conformal field theory has central charge $c=0$. The prevailing folklore — the lattice structure is irrelevant, so both models lie in the same universality class — is hard to formulate sharply because a $c=0$ conformal field theory is logarithmic (LCFT): its representations are reducible yet indecomposable, and the Virasoro generator $L_0$ (the scaling Hamiltonian, whose eigenvalues are the conformal weights) acts non-diagonalizably, with Jordan cells tying together distinct states of equal conformal weight. The off-diagonal matrix elements, or logarithmic couplings, control logarithmic corrections to correlation functions. Two LCFTs can therefore share all conformal weights and critical exponents yet differ in their Jordan-cell structure, so 'same universality class' is for logarithmic theories a strictly stronger statement than equality of spectra.

Both percolation problems have Yang–Baxter integrable loop-model regularizations. A loop model is a lattice model whose configurations are collections of non-intersecting loops, each closed contractible loop carrying a fugacity $\beta$; its local move algebra is the Temperley–Lieb (TL) diagram algebra, whose 'dilute' variant additionally allows sites to remain vacant. Critical bond percolation on the square lattice is the dense $A_1^{(1)}$ loop model at $\beta=1$; critical site percolation on the triangular lattice is the dilute $A_2^{(2)}$ loop model — built from the Izergin–Korepin nineteen-vertex solution of the Yang–Baxter equation (the consistency condition making the row-to-row transfer matrices commute) — at crossing parameter $\lambda=\pi/3$, where $\beta=-2\cos 4\lambda=1$. At root-of-unity values these become the logarithmic minimal models $\mathcal{LM}(p,p')$ and $\mathcal{DLM}(p,p')$ with $c=1-6(p-p')^2/(pp')$ — the dense family at $\lambda/\pi=(p'-p)/p'$ and the dilute family at $\lambda/\pi=(2p'-p)/(4p')$, the two relations coinciding at percolation, $(p,p')=(2,3)$, $\lambda=\pi/3$. On strips and tori the transfer matrices act on standard modules of the (periodic) dilute TL algebra, labelled by the number $d$ of defects (through-lines) and, on the torus, by the twist angle $\gamma$ ($\omega=e^{i\gamma}$); a Markov trace (a natural weighted trace on the diagram algebra) over these modules, depending on the fugacity $\alpha$ of non-contractible loops, produces the cylinder and torus partition functions of the scaling-limit theory.

The functional-relations program solves such models exactly. The commuting transfer matrices, fused according to the nodes $(m,n)$ of the dominant integral weight lattice of $s\ell(3)$ (the Lie algebra of traceless $3\times 3$ matrices, whose dominant integral weights are pairs of nonnegative integers $(m,n)$), satisfy a fusion hierarchy that closes into finite $T$- and $Y$-systems at roots of unity; Morin-Duchesne and Pearce (2019) derived these for the dilute $A_2^{(2)}$ loop model and noted that the associated Thermodynamic Bethe Ansatz (TBA) and nonlinear integral equations (NLIE) determine the finite-size $1/N$ corrections that fix the central charge and conformal weights. This program has since been carried out: ground-state dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$, $A_2^{(2)}$ models (2021); the full NLIE solution at $\lambda=\pi/3$ (2023), giving $c=0$, the ground-state conformal weights analytically, conjectural scaling-limit traces in every standard module, and the conformal cylinder and torus partition functions — which coincide precisely with the earlier bond-percolation results obtained from $D_3$ TBA equations; and the extension to all $(p,p')$ (2025), where the dense and dilute conformal partition functions are identical for every pair $(p,p')$, a coincidence that also appears to rule out identifying critical percolation with the $c=0$ triplet model $\mathcal{WLM}(2,3)$ (a different candidate logarithmic theory at $c=0$, of 'triplet' type). All of this evidence, however, lives at the level of spectra — the characters entering the partition functions — and does not touch the Jordan cells.

Jordan cells in lattice models are accessible in two regimes. At roots of unity the finite-size transfer matrices can themselves be defective: for the dilute $A_2^{(2)}$ model at the percolation point, the single-row transfer matrix at $\lambda=\pi/3$ 'can exhibit nontrivial Jordan blocks', in which case there is no complete set of eigenvectors and the matrix is not diagonalizable. For the dense percolation lineage, Morin-Duchesne and Saint-Aubin directly mapped the Jordan cells of the periodic TL loop models (including the bond-percolation point); the corresponding analysis for the dilute $A_2^{(2)}$ percolation model and its periodic dilute TL algebra has not been performed, and the general emerging-block technique recently developed by Liu, Jacobsen and Saleur for the Potts model at generic $Q$ — extracting scaling-limit cells from nearly degenerate eigenvalue pairs of diagonalizable finite-size matrices — has not been applied to this algebra. On the continuum side, Camia and Feng have proven the emergence of logarithmic singularities in the scaling limit of critical percolation, but a full percolation LCFT — in particular its indecomposable Virasoro module structure — remains open. The question raised by the 2019 paper and restated as the next step in its 2023 solution is thus precise: is the Jordan-cell structure of the site-percolation scaling limit the same as that of bond percolation?

Problem Statement

Determine the Jordan-cell structure of the scaling-limit logarithmic conformal field theory of the dilute $A_2^{(2)}$ loop model at the percolation point $\lambda=\pi/3$ (critical site percolation on the triangular lattice, $\mathcal{DLM}(2,3)$), and thereby decide whether bond and site percolation share the same indecomposable Virasoro structure.

Concretely, for each standard module of the periodic dilute Temperley–Lieb algebra at $\lambda=\pi/3$ — labelled by the defect number $d$ and the twist $\gamma$ — and for each value of the non-contractible loop fugacity $\alpha$ entering the Markov trace, an answer should determine, in the scaling limit: (i) which conformal weights are coupled in Jordan cells of $L_0$, and the ranks of those cells; and (ii) the associated logarithmic couplings. It should then compare this structure, sector by sector, with the Jordan-cell structure established for the dense $A_1^{(1)}$ loop model at $\beta=1$ (critical bond percolation, $\mathcal{LM}(2,3)$) and state the verdict: either the cells and logarithmic couplings agree in every sector, establishing that bond and site percolation lie in the same universality class as logarithmic conformal field theories in the strong sense that goes beyond the already-established equality of spectra and conformal partition functions; or they differ in some sector, which refutes that strong universality, in which case the differing cells and logarithmic couplings must be specified in every sector where the two structures differ.

The common objective uniting these deliverables is a decision on the strong universality of the $c=0$ percolation logarithmic CFT across its two canonical integrable regularizations; the Jordan-cell structure and logarithmic couplings must be determined in every standard module (all $d$, $\gamma$ and $\alpha$), and results that cover only some sectors, or that stop short of the comparison, do not complete it.

References

1. Alexi Morin-Duchesne, Paul A. Pearce, Fusion hierarchies, T-systems and Y-systems for the dilute $A_2^{(2)}$ loop models, J. Stat. Mech. (2019) 094007, doi:10.1088/1742-5468/ab3412, arXiv:1905.07973, https://arxiv.org/abs/1905.07973
2. Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce, Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models, J. Stat. Mech. (2021) 033105, doi:10.1088/1742-5468/abdc17, arXiv:2006.08233, https://arxiv.org/abs/2006.08233
3. Florence Boileau, Alexi Morin-Duchesne, Yvan Saint-Aubin, Fusion hierarchies, T-systems and Y-systems for the dilute $A_2^{(2)}$ loop models on a strip, J. Stat. Mech. (2023) 033102, doi:10.1088/1742-5468/acb7d7, arXiv:2211.09017, https://arxiv.org/abs/2211.09017
4. Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce, Critical site percolation on the triangular lattice: from integrability to conformal partition functions, J. Stat. Mech. (2023) 043103, doi:10.1088/1742-5468/acc4b3, arXiv:2211.12379, https://arxiv.org/abs/2211.12379
5. Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce, Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models, J. Stat. Mech. (2025) 073103, doi:10.1088/1742-5468/ade133, arXiv:2501.19288, https://arxiv.org/abs/2501.19288
6. Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce, Conformal partition functions of critical percolation from $D_3$ thermodynamic Bethe Ansatz equations, J. Stat. Mech. (2017) 083101, doi:10.1088/1742-5468/aa75e2, arXiv:1701.08167, https://arxiv.org/abs/1701.08167
7. Alexi Morin-Duchesne, Yvan Saint-Aubin, Jordan cells of periodic loop models, J. Phys. A 46 (2013) 494013, doi:10.1088/1751-8113/46/49/494013, arXiv:1302.5483, https://arxiv.org/abs/1302.5483
8. Lawrence Liu, Jesper Lykke Jacobsen, Hubert Saleur, Emerging Jordan blocks in the two-dimensional Potts and loop models at generic Q, JHEP 02 (2025) 080, doi:10.1007/JHEP02(2025)080, arXiv:2403.19830, https://arxiv.org/abs/2403.19830
9. Federico Camia, Yu Feng, Conformally covariant probabilities, operator product expansions, and logarithmic correlations in two-dimensional critical percolation, arXiv:2407.04246, https://arxiv.org/abs/2407.04246
10. Jonathan Belletête, Yvan Saint-Aubin, The principal indecomposable modules of the dilute Temperley-Lieb algebra, J. Math. Phys. 55 (2014) 111706, doi:10.1063/1.4901546, arXiv:1310.4791, https://arxiv.org/abs/1310.4791
11. Alexis Langlois-Rémillard, Alexi Morin-Duchesne, Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors, Commun. Algebra (2026), doi:10.1080/00927872.2026.2647189, arXiv:2302.12782, https://arxiv.org/abs/2302.12782