Trace conjectures and zero-pattern classification for the dilute $A_2^{(2)}$ loop model at $λ=π/3$: the conformal partition functions of critical site percolation on the triangular lattice

Critical site percolation on the triangular lattice is described exactly by the Yang–Baxter-integrable dilute $A_2^{(2)}$ loop model with crossing parameter $λ=π/3$ (contractible loop weight $β=-2\cos 4λ=1$). Morin-Duchesne, Klümper and Pearce (2023) derived the model's functional relations — a fusion hierarchy closing into cubic functional equations, Baxter $T$–$Q$ relations and Bethe ansatz equations involving auxiliary functions $Q(u)$ and $P(u)$ — and solved the resulting nonlinear integral equations (NLIEs) analytically for the ground-state conformal weights in every standard module. However, they could not classify the analytic zero patterns of the eigenvalues and auxiliary functions, an $sℓ(3)$-level obstruction absent from their completed $sℓ(2)$ analysis of bond percolation. Their Conjectures 1 and 2, giving the scaling limits of the transfer-matrix traces in all standard modules — and hence, through the Markov trace, the conformal partition functions of site percolation on the cylinder and torus — are supported only by high-precision numerics for $d≤8$. The problem posed here is to establish (or refute) these trace formulas for all defect numbers $d$, twists $γ$, and excitations. The natural integrability route to these formulas entails classifying the analytic zero patterns (strings, holes, zeros at infinity) for arbitrary eigenstates and system sizes, constructing NLIEs — or an equivalent $Y$-system/thermodynamic-Bethe-ansatz route, possibly of Kuniba–Sakai–Suzuki 'snake' type — with the correct analyticity properties for all excitations, and evaluating the resulting dilogarithm integrals to obtain the character formulas. A 2025 follow-up by the same authors extended the same conjectures to all root-of-unity dense and dilute logarithmic minimal models without proving them.

Background

Site percolation on the triangular lattice — each site occupied independently with probability $p$ — is critical at $p=1/2$, and its continuum limit is among the best-understood two-dimensional critical points: it is conformally invariant in the sense of Schramm–Loewner evolution and is described by a logarithmic conformal field theory (CFT) with central charge $c=0$. 'Logarithmic' means that the Virasoro symmetry algebra acts through indecomposable but reducible representations, so correlation functions may contain logarithms — as percolation hull observables do — making the representation theory far richer than that of ordinary rational CFTs.

Exact lattice calculations proceed through a loop representation. Following Fehér and Nienhuis, each configuration of critical site percolation on the triangular lattice maps to a configuration of a dilute loop model on the square lattice whose loop segments trace the domain walls separating occupied from empty clusters. This loop model is Yang–Baxter integrable: it is the dilute $A_2^{(2)}$ loop model, built on the Izergin–Korepin nineteen-vertex $R$-matrix (the rank-2 solvable vertex-model Boltzmann-weight matrix acting on three-state spin spaces), with crossing parameter $λ$ and contractible loop weight $β=-2\cos 4λ$; site percolation corresponds to the root of unity $λ=π/3$, where $β=1$.

The model is analyzed through commuting transfer matrices acting on representations of diagram algebras. The Temperley–Lieb algebra is generated by cup-and-cap diagrams modulo relations in which every closed loop evaluates to the weight $β$; its dilute versions also allow strand configurations relevant to a dilute loop gas. The relevant representations are the standard modules $W_{N,d,ω}$ of the periodic dilute Temperley–Lieb algebra and $V_{N,d}$ of the ordinary one, labelled by the number $d$ of defects (strands passing through the system) and, in the periodic case, by a twist $ω=e^{iγ}$ accumulated by such strands. The single-row (periodic) and double-row (strip) transfer matrices have eigenvalues satisfying functional relations: a fusion hierarchy — the tower of transfer matrices obtained by fusing the fundamental one, tied together by functional equations — that at $λ=π/3$ closes into cubic functional relations, together with Baxter $T$–$Q$ relations (functional equations expressing the transfer-matrix eigenvalue through an auxiliary function $Q(u)$) and Bethe ansatz equations involving two auxiliary functions $Q(u)$ and $P(u)$.

The standard route from such relations to conformal data is the nonlinear integral equation (NLIE) method. One classifies the analytic structure of the auxiliary functions in the complex spectral plane — the 'pattern of zeros', organized into strings (regularly spaced Bethe roots: here 1-, 2- and 3-strings), holes (vacant positions) and zeros at infinity — counts these objects for each eigenstate, and converts the functional equations into a finite set of NLIEs. Solving them in the scaling limit and evaluating the resulting dilogarithm integrals yields the finite-size corrections of the transfer-matrix eigenvalues, hence the central charge and conformal weights $Δ_{r,s}=((3r-2s)^2-1)/24$ of the Kac table. For $sℓ(2)$-type models this program was carried through: for critical bond percolation on the square lattice (the dense $A_1^{(1)}$ loop model at $λ=π/3$), Morin-Duchesne, Klümper and Pearce conjectured a complete classification of the patterns of zeros of all eigenvalues — supported by extensive numerics and used as selection rules — and, taking it as input, rewrote the resulting $Y$-system as $D_3$-type thermodynamic Bethe ansatz (TBA) equations and derived the conformal partition functions. For the rank-2 $sℓ(3)$ dilute $A_2^{(2)}$ model, the cubic functional relations entangle $Q$ and $P$ in a way that has so far resisted an analogous classification.

The objects of interest are the traces of the transfer-matrix powers over the standard modules. The dilute Temperley–Lieb algebras carry Markov traces — weighted traces in which closed contractible and non-contractible loops receive weights $β$ and $α$ — so the scaling limits of these traces simultaneously encode the conformal partition functions of site percolation on the torus (single-row case) and on the cylinder (double-row case), expressible as sesquilinear forms (bilinear up to complex conjugation of the second argument) in Virasoro Verma characters $χ(q)=\mathrm{tr}\,q^{L_0-c/24}$, or, at loop fugacity $α=2$, in affine $u(1)$ characters.

In the 2023 analysis the NLIEs were solved analytically only for the ground states of the standard modules, giving $Δ=Δ_{1,d+1}$ on the strip and $(Δ,Δ̄)=(Δ_{γ/π,d/2},Δ_{γ/π,-d/2})$ in the periodic case (assuming the known $c=0$); excited states were reached through high-precision numerical solutions of the logarithmic Bethe ansatz equations. The authors also note that the known $A_2^{(2)}$ $Y$-system — the algebraic system satisfied by auxiliary functions, of Gliozzi–Tateo type tied to ADE functional dilogarithm identities — involves functions lacking the analytic-nonzero-constant-asymptotics properties needed for a TBA computation of conformal eigenenergies, and that the generalized 'snake' $Y$-systems constructed by Kuniba, Sakai and Suzuki for the six-vertex (XXZ) model at roots of unity have no known $A_2^{(2)}$ analogue. Remarkably, the conjectured conformal data coincide exactly with those previously found for bond percolation — evidence for a strong form of universality between the two lattice realizations as logarithmic CFTs, which a proof of the trace conjectures would place on analytic footing.

Problem Statement

Establish — or refute — Conjectures 1 and 2 of Morin-Duchesne, Klümper and Pearce for the dilute $A_2^{(2)}$ loop model at $λ=π/3$, i.e. for critical site percolation on the triangular lattice, in all standard modules and for all excitations.

Conjecture 1 (periodic sector; traces of the single-row transfer matrix) asserts that, with $ω=e^{iγ}$, $q=\exp(-2πiδ e^{-iu})$, $q̄=\exp(2πiδ e^{iu})$ and $(q)_∞=\prod_{k≥1}(1-q^k)$,

$$\lim_{M,N\to\infty,\; M/N=δ} e^{MN f_{\mathrm{bulk}}(u)}\,\mathrm{tr}_{W_{N,d,ω}}\, \widehat T(u)^M=\sum_{ℓ=-∞}^{∞}\frac{q^{Δ_{γ/π-2ℓ,\,d/2}}\,q̄^{Δ_{γ/π-2ℓ,\,-d/2}}}{(q)_∞(q̄)_∞}$$

for every standard module $W_{N,d,ω}$ of the periodic dilute Temperley–Lieb algebra. Conjecture 2 (strip sector; traces of the double-row transfer matrix) asserts

$$\lim_{M,N\to\infty,\; M/N=δ} e^{2MN f_{\mathrm{bulk}}(u)+M f_{\mathrm{bdy}}(u)}\,\mathrm{tr}_{V_{N,d}}\, \widehat D(u)^M=q^{Δ_{1,d+1}}\,\frac{1-q^{d+1}}{(q)_∞},\qquad q=\exp(-2πδ\sin u).$$

Here $\widehat T(u)$ and $\widehat D(u)$ are the suitably normalized single- and double-row transfer matrices of the source paper — the normalization under which its eigenvalue expansions (4.2) hold — $u$ is the spectral parameter, $δ$ is the aspect ratio, and the bulk and boundary free energies are

$$f_{\mathrm{bulk}}(u)=-\log\left[\tfrac{4}{\sqrt{3}}\,\sin\tfrac{1}{2}\!\left(u+\tfrac{\pi}{3}\right)\sin\tfrac{1}{2}\!\left(u+\tfrac{2\pi}{3}\right)\right],\qquad f_{\mathrm{bdy}}(u)=0.$$

The known route to these formulas — the one the source authors could not complete, in contrast with their $sℓ(2)$ bond-percolation analysis, where a conjectural classification of the patterns of zeros of all eigenvalues (supported by extensive numerics) was available and served as selection rules — runs through a classification of the analytic zero patterns (1-, 2- and 3-strings, holes and zeros at infinity) of the transfer-matrix eigenvalues and of the auxiliary functions $Q(u)$ and $P(u)$ for arbitrary eigenstates, defect numbers $d$, twists $γ$ and system sizes $N$, in a form that survives the scaling limit; the derivation from such a classification of nonlinear integral equations — whether by the extended functional-relation route of the source paper or through an equivalent $Y$-system/thermodynamic-Bethe-ansatz construction (for instance a Kuniba–Sakai–Suzuki-type 'snake' $Y$-system for $A_2^{(2)}$) — whose auxiliary functions possess the analyticity properties needed to describe arbitrary excitations, not only ground states; and the solution of these equations in the scaling limit with exact evaluation of the corresponding dilogarithm integrals, yielding the trace formulas above and, via the Markov trace, the conformal partition functions of critical site percolation on the cylinder and torus, including the modular-invariant torus partition function. This route is the known one, not a mandatory one: the target of the problem is the truth of Conjectures 1 and 2 themselves, however it is established.

The already-proven ground-state leading terms do not count towards completion: the target is the full set of excitations and the complete trace formulas. A proof of the more general root-of-unity trace conjecture formulated by the same authors in 2025, which extends Conjecture 1 to all dense $\mathrm{LM}(p,p')$ and dilute $\mathrm{DLM}(p,p')$ loop models on the torus, would resolve the periodic-sector part of this problem, but not the strip-sector Conjecture 2, which that extension does not cover. A refutation consists in exhibiting a standard module and eigenstate whose scaling-limit trace contradicts the conjectured formula.

References

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