Wu's 2006 Physical Review Letters paper derived closed-form critical frontiers for qq-state Potts models and their percolation limits on lattices decomposable into three-terminal triangular units — the martini, martini-A, and martini-B lattices and related graphs — by identifying the critical surface with the self-dual trajectory qA−C=0qA-C=0 of the Baxter–Temperley–Ashley duality, an identification that is rigorous only in the ferromagnetic parameter regime. Later work rigorized the ferromagnetic triangular-type case, extended exact frontiers to bow-tie-type lattices, and produced two families of conjectural frontiers for the kagome lattice — Wu's homogeneity-assumption closed form and the critical polynomials of Jacobsen and Scullard — whose numerical inexactness is now demonstrated (deviations of order 10−510^{-5} to 10−410^{-4} from high-precision determinations, and a rigorous disproof of an older conjectured kagome bond-percolation threshold). The precise open core that survives: the exact Potts critical manifold of the kagome lattice and other planar lattices lacking the triangular decomposition, critical frontiers for antiferromagnetic or sign-indefinite couplings on the decorated lattices, a rigorous justification or refutation of self-dual-locus criticality beyond the ferromagnetic regime, and a systematic classification of network realizations admitting closed-form frontiers.

Background

The qq-state Potts model places a spin σi∈{1,…,q}\sigma_i\in\{1,\dots,q\} on every site of a lattice, with a Boltzmann weight exp⁡(K δσi,σj)\exp(K\,\delta_{\sigma_i,\sigma_j}) that favours equal values on neighbouring spins (δ\delta is the Kronecker delta). Its critical frontier is the locus in coupling space separating the ordered low-temperature phase from the disordered high-temperature phase. In two dimensions the frontier is exactly known only for a handful of lattices: the square lattice (by Kramers–Wannier self-duality, the exact equivalence of a planar lattice model with a model on the dual lattice) and the triangular and honeycomb lattices (by the star–triangle relation, the exact equivalence of a triangle of pairwise couplings with a three-legged star). For q=1q=1 with two-site interactions the Potts partition function reduces to that of bond percolation — the study of global connectivity of independently occupied bonds — while versions with multi-site (three-spin) interactions generate site and mixed site–bond percolation; the Fortuin–Kasteleyn random-cluster representation, an expansion of the Potts partition function as a sum over bond configurations, makes this correspondence systematic. Exact Potts frontiers therefore double as exact percolation thresholds.

A broader framework dates to Baxter, Temperley and Ashley (1978). Build a lattice from triangular units: each unit is an arbitrary network of spins that connects to the rest of the lattice only through three terminal spins σ1,σ2,σ3\sigma_1,\sigma_2,\sigma_3, so that its effective Boltzmann factor can be written

F(σ1,σ2,σ3)=A+B1δ23+B2δ31+B3δ12+C δ123,F(\sigma_1,\sigma_2,\sigma_3)=A+B_1\delta_{23}+B_2\delta_{31}+B_3\delta_{12}+C\,\delta_{123},

where δ123=δ12δ23\delta_{123}=\delta_{12}\delta_{23} signals all three spins equal. Such a model obeys a duality relation in the parameter space (A,B1,B2,B3,C)(A,B_1,B_2,B_3,C) whose self-dual trajectory — the set of parameters mapped to itself under duality — is the hypersurface qA−C=0qA-C=0. Wu and Lin (1980) gave a graphical proof of this duality, and Wu and Zia (1981) proved rigorously that within the ferromagnetic regime, the parameter region satisfying B1+B2+B3+C>0B_1+B_2+B_3+C>0 and Bi+Bj+C>0B_i+B_j+C>0 for i≠ji\ne j, the critical frontier is exactly this self-dual trajectory. Chayes and Lei later proved rigorous random-cluster theorems for the triangular-lattice version with two- and three-site interactions.

Scullard and Ziff (2005–2006) used star-triangle and cell–dual-cell transformations to obtain exact site- and bond-percolation thresholds on the martini lattices — decorated honeycomb lattices in which alternate sites are replaced by triangles. Wu's 2006 Physical Review Letters paper then showed that the Baxter–Temperley–Ashley machinery gives, for any realization of the triangular-unit structure with two- and three-site interactions, an explicit algebraic Potts critical frontier: compute the unit's parameters (A,Bi,C)(A,B_i,C) and impose qA−C=0qA-C=0. He worked out closed forms for the martini, martini-A and martini-B lattices and a further decorated lattice, recovering the Scullard–Ziff percolation thresholds as q=1q=1 limits and adding new exact thresholds.

What the 2006 construction does not do defines the present problem. First, the identification of the self-dual trajectory with the physical critical surface is proved only in the ferromagnetic regime; antiferromagnetic or sign-indefinite couplings — where the monotonicity underlying random-cluster arguments fails — are not covered. Second, the lattice must decompose into triangular units of a single orientation glued at their terminals; the kagome lattice (the Archimedean (3,6,3,6)(3,6,3,6) lattice, in which every vertex touches two triangles and two hexagons, i.e. the lattice of corner-sharing triangles) does not have this form, and its Potts critical frontier has been sought since the early 1980s. Third, only particular network realizations were treated, with no systematic classification of which units yield algebraically tractable frontiers. Approximation tools fill part of the gap — notably the critical polynomials PB(q,v)P_B(q,v) of Jacobsen and Scullard, algebraic curves computed from a finite repeating basis BB of the lattice that converge numerically toward the true critical manifold — but their exactness is conjectural, resting on an unproven infinite-basis limit.

Problem Statement

What is settled (Wu, 2006, building on Baxter–Temperley–Ashley and Wu–Zia): for ferromagnetic Potts models with two- and three-site interactions on lattices decomposable into single-orientation three-terminal triangular units, the critical frontier is exactly the self-dual trajectory qA−C=0qA-C=0. The open problem is to carry exact, provably exact critical frontiers beyond that setting. Facet (1) is the central question — the determination of the Potts critical frontier for lattices outside the solved class, paradigmatically the kagome lattice — and resolving it suffices to close the problem; facets (2)–(4) are optional further directions in the same scope. All share one scientific objective — an exact determination of Potts and percolation critical frontiers on planar lattices:

(1) Non-decomposable lattices. Determine the exact critical frontier of the qq-state Potts model, including the q=1q=1 bond-percolation limit, on lattices that cannot be built from single-orientation three-terminal triangular units — paradigmatically the kagome lattice and its decorated kagome-type relatives. Establishing the exact frontier for the kagome lattice alone suffices to resolve facet (1); the wider class of non-decomposable lattices is the motivating scope. An acceptable answer establishes the exact algebraic manifold together with a proof of exactness (not a numerical fit).

(2) Non-ferromagnetic regimes. Locate the critical frontier of the triangular-decomposition lattices (the martini family and its generalizations, with inhomogeneous two- and three-site interactions) for antiferromagnetic or sign-indefinite couplings, where the Wu–Zia ferromagnetic proof does not apply.

(3) Rigour of self-duality. Prove or refute, outside the ferromagnetic regime and for the most general inhomogeneous multi-site interactions on the admissible lattices, that the self-dual trajectory qA−C=0qA-C=0 coincides with the physical critical surface.

(4) Classification. Characterize systematically which triangular-network realizations have parameters (A,Bi,C)(A,B_i,C) reducible to explicit polynomials in the couplings, and hence admit closed-form frontiers, for example along the lines of the self-dual planar hypergraph class of Wierman and Ziff (2011).

Results confined to the ferromagnetic triangular-decomposition class, or numerical accuracy without proof of exactness, do not close the problem.

References

  1. F. Y. Wu, "New critical frontiers for the Potts and percolation models", Physical Review Letters 96, 090602 (2006). DOI: 10.1103/PhysRevLett.96.090602, arXiv:cond-mat/0601150. https://doi.org/10.1103/PhysRevLett.96.090602
  2. R. J. Baxter, H. N. V. Temperley, S. E. Ashley, "Triangular Potts model at its transition temperature, and related models", Proceedings of the Royal Society of London A 358, 535–559 (1978). DOI: 10.1098/rspa.1978.0026. https://doi.org/10.1098/rspa.1978.0026
  3. F. Y. Wu, K. Y. Lin, "On the triangular Potts model with two- and three-site interactions", Journal of Physics A 13, 629–636 (1980). DOI: 10.1088/0305-4470/13/2/026. https://doi.org/10.1088/0305-4470/13/2/026
  4. F. Y. Wu, R. K. P. Zia, "Critical point of a triangular Potts model with two- and three-site interactions", Journal of Physics A 14, 721–727 (1981). DOI: 10.1088/0305-4470/14/3/018. https://doi.org/10.1088/0305-4470/14/3/018
  5. L. Chayes, H. K. Lei, "Random Cluster Models on the Triangular Lattice", arXiv:cond-mat/0508254 (2005). https://arxiv.org/abs/cond-mat/0508254
  6. C. R. Scullard, "Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation", Physical Review E 73, 016107 (2006). DOI: 10.1103/PhysRevE.73.016107, arXiv:cond-mat/0507392. https://doi.org/10.1103/PhysRevE.73.016107
  7. R. M. Ziff, "Generalized cell–dual-cell transformation and exact thresholds for percolation", Physical Review E 73, 016134 (2006). DOI: 10.1103/PhysRevE.73.016134, arXiv:cond-mat/0510245. https://doi.org/10.1103/PhysRevE.73.016134
  8. F. Y. Wu, "Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions", Physical Review E 81, 061110 (2010). DOI: 10.1103/PhysRevE.81.061110, arXiv:0911.2514. https://doi.org/10.1103/PhysRevE.81.061110
  9. C. Ding, Z. Fu, W. Guo, F. Y. Wu, "Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis", Physical Review E 81, 061111 (2010). DOI: 10.1103/PhysRevE.81.061111, arXiv:1001.1488. https://doi.org/10.1103/PhysRevE.81.061111
  10. C. R. Scullard, R. M. Ziff, "Critical surfaces for general inhomogeneous bond percolation problems", Journal of Statistical Mechanics P03021 (2010). DOI: 10.1088/1742-5468/2010/03/P03021, arXiv:0911.2686. https://doi.org/10.1088/1742-5468/2010/03/P03021
  11. S. Baek, H. Mäkelä, P. Minnhagen, B. J. Kim, "Critical temperatures of the three- and four-state Potts models on the kagome lattice", Physical Review E 83, 061104 (2011). DOI: 10.1103/PhysRevE.83.061104, arXiv:1106.1296. https://doi.org/10.1103/PhysRevE.83.061104
  12. J. C. Wierman, R. M. Ziff, "Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds", Electronic Journal of Combinatorics 18 (2011). DOI: 10.37236/548, arXiv:0903.3135. https://doi.org/10.37236/548
  13. B. Bollobás, O. Riordan, "Percolation on Self-Dual Polygon Configurations", Bolyai Society Mathematical Studies 21, 131–217 (2010). DOI: 10.1007/978-3-642-14444-8_3. https://doi.org/10.1007/978-3-642-14444-8_3
  14. J. L. Jacobsen, C. R. Scullard, "Critical manifold of the kagome-lattice Potts model", Journal of Physics A 45, 494003 (2012). DOI: 10.1088/1751-8113/45/49/494003, arXiv:1204.0622. https://doi.org/10.1088/1751-8113/45/49/494003
  15. J. L. Jacobsen, C. R. Scullard, "Transfer matrix computation of critical polynomials for two-dimensional Potts models", Journal of Physics A 46, 075001 (2013). DOI: 10.1088/1751-8113/46/7/075001, arXiv:1211.4335. https://doi.org/10.1088/1751-8113/46/7/075001
  16. C. R. Scullard, J. L. Jacobsen, "Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices", Journal of Physics A 46, 125006 (2013). DOI: 10.1088/1751-8113/46/12/125006, arXiv:1210.7845. https://doi.org/10.1088/1751-8113/46/12/125006
  17. J. L. Jacobsen, "High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials", Journal of Physics A 47, 135001 (2014). DOI: 10.1088/1751-8113/47/13/135001, arXiv:1401.7847. https://doi.org/10.1088/1751-8113/47/13/135001
  18. C. Ding, Y. Wang, Y. Li, "Potts and percolation models on bowtie lattices", Physical Review E 86, 021125 (2012). DOI: 10.1103/PhysRevE.86.021125, arXiv:1203.2244. https://doi.org/10.1103/PhysRevE.86.021125
  19. C. Ding, Z. Fu, W. Guo, "Critical points of the O(n) loop model on the martini and the 3-12 lattices", Physical Review E 85, 062101 (2012). DOI: 10.1103/PhysRevE.85.062101, arXiv:1202.5623. https://doi.org/10.1103/PhysRevE.85.062101
  20. J. C. Wierman, G. Yu, T. Huang, "A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice", Electronic Journal of Combinatorics 22(2), P2.52 (2015). DOI: 10.37236/5117. https://doi.org/10.37236/5117
  21. H. Duminil-Copin, V. Sidoravicius, V. Tassion, "Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1 ≤ q ≤ 4", Communications in Mathematical Physics 349, 47–107 (2016). DOI: 10.1007/s00220-016-2759-8. https://doi.org/10.1007/s00220-016-2759-8
  22. H. Duminil-Copin, M. Gagnebin, M. Harel, I. Manolescu, V. Tassion, "Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4", Annales scientifiques de l'École Normale Supérieure 54, 1363–1413 (2021). DOI: 10.24033/asens.2485. https://doi.org/10.24033/asens.2485
  23. C. R. Scullard, J. L. Jacobsen, R. M. Ziff, "Critical percolation on the kagome hypergraph", Journal of Physics A 54, 055006 (2021). DOI: 10.1088/1751-8121/abcddb. https://doi.org/10.1088/1751-8121/abcddb